Which of the U.S. national parks in this image do you think is the most worth visiting? There are three exceptions.

Shenandoah and the Great Smokey mountains.
Drive the blue ridge parkway. it's about 4 days with plenty of great sights off the side of the road.
- This! It's beautiful in late summer, early fall. Lots of great easy and moderate hiking. Hit up Front Royal, VA for some food and booze.
- Definitely this. Great areas.

Breaks interstate Park between WV and KY is pretty nice. I also very, very highly recommend smoky mountains, I live in the region and have gone every year. Ik you said no cities but Gatlinburg area can be nice to check out after the trip if you wanna see Smoky Mountain Knife Works and all the small businesses in the rural parts of the mountains.
- Thanks, this is the sort of advice I was hoping for!
  
  +1 for Smoky Mtns. So much to explore, Clingmans Dome, waterfalls, elk, bears, restaurants close by, but you can get as far away from the tourist-ty stuff as you like. Secluded cabins to downtown hotels. Dollywood for the kids is good. I live in Georgia, but try to get up there with the family every year or two. Also Smoky mountain knife works is worth it if you're into knives, outdoors, camping...
  
  The Artisan's Market is worth going to Gatlinburg, but avoid the main strip of the city unless you're looking to eat. There's maybe 4 unique shops, the rest are the exact same touristy shit under a different name.
  The Knife Works will be busy, also. It doesn't slow down. Parking will be ass. It's totally worth it just to go into the downstairs Relic Room.

If you're willing to go as far as Kentucky or West Virginia anyway, you should consider the Red River Gorge area in Daniel Boone in KY, or the Spruce Knob/Seneca Rocks area in WV. Neither are national parks -- they're both national forests.
Both will be considerably less touristy and less crowded than (at least the popular) national parks, and you don't have to pay just to get in, either. These two areas have some of the most bodacious geology on display on the East Coast, in my opinion, and if you're into that sort of thing it's well worth checking out.
The heyday of the Seneca Rocks region seems to have passed and getting accommodation there that's not camping is trickier than it was a few decades ago, since most of the motels and hotels around the region have folded. But you can rent cabins if you plan in advance from various outfits, and there are two quite nice national forest camp sites there plus oodles of commercial/independent ones. Seneca Rocks itself is a quite striking geological feature you can hike up and stand on top of, and Spruce Knob is just a hop, skip, and a jump away and is the highest point in WV with some great and very easily accessible views from the top. Don't forget to stop by Yocum's general store and pet the cats when you're there.
Dispersed camping is no longer allowed within the Seneca Rocks/Spruce Knob sphere of influence, but it is in the rest of the adjacent greater Monongahela National Forest, including in the Dolly Sods wilderness if you're into that sort of thing. Backpacking in Dolly Sods is quite possibly the best way to see the most varied terrain anywhere east of the Mississippi within the span of a weekend and without owning a private jet. The north, east, south, and west extremities of it may as well be on different continents; it's pretty wild.
Dispersed camping is allowed in Daniel Boone if you go there. You have to buy a permit to leave your car anywhere overnight to go backpacking but it's only a couple of bucks. The Red River Gorge area in Daniel Boone has some incredible sandstone formations including massive arches (some of which you can climb), shelter caves, cliffs, and overlooks. It's also home to the Nada Tunnel which is pretty cool but maybe not so appealing to people who are afraid of caves because it's basically a cave with a one lane road you can drive straight through. (It was actually originally a railway tunnel. I cannot possibly conceive of what it must have been like to cram a coal burning steam locomotive through that tiny passage, and if you see it you'll know why. But that's what they did back in the day.)
Civilized accommodations are easier to come by there including plenty of cabins and motels, and also hotels you can find near the interstate. If you're into rock climbing there are also a ton of climbing routes all over the Red River Gorge.
Forget Yellowstone. Yellowstone is so popular and yet so fragile and so dangerous that the entire place is on lockdown. You spend your entire stay there on rails, pretty much literally. Everything is boardwalks and pavement and everything else that isn't is cordoned off. Yes, this is so dumbasses cannot fall into geothermal features and be boiled alive. But it also has the net effect of causing you to take the exact same route in the exact same way and take the exact same pictures that everyone else already has. So you can have the same experience by just finding some rando's Flickr album or whatever and looking at their pictures, because they'll be just the same as yours. Plus the whole place stinks. Sulfur, don't you know.
Oh, and you get to contend with access roads clogged by all the dimwits from the midwest who stop dead in the middle of everything to try to fit baby bison into their minivans, or whatever the fuck else. I went once and that was enough. I came, I saw, I bought a mug. I have no desire to go back. (Nearby Shoshone National Forest, however, is friggin' awesome. So is Big Horn. Be sure to check out Shell Falls while you're there and annoy your nearest creationist.)
- Thanks. I'm doing some research now. Daniel Boone we could probably do over a long weekend, so that could be a separate trip. Seneca Rocks looks really beautiful in photos, but I'm not convinced there would be enough to do there to sustain a week's vacation. As far as a cabin, my daughter always balks at renting one when we've suggested in the past for some reason. I don't know why. We did it at a nearby state park when she was younger and it was fine, so I don't know what her deal is there.
  
  If you wanted to extend a Red River Gorge trip there's some interesting Civil War era stuff in Winchester and Lexington. Fort Boonesborough was rebuilt as a Civil War fort and they do history presentations and era accurate crafting demonstrations. They have a working blacksmith, a soap maker, that kind of thing. The Henry Clay Estate is interesting and the Cassius Clay Estate (the abolitionist, General, and Diplomat not the boxer) is great. There's also the Kentucky Horse Park and Keeneland. You will also be passing through the Bourbon Trail if you're driving down from Indiana and by Big Bone Lick if you're coming down 65.
- As someone that has been to the tops of Cloud Splitter, Grays Arch, Chimney Rock, Half Moon, Indian Staircase, and dozens of others I cannot remember at the moment, Red River Gorge is the single best place in all of Kentucky.
  I would go as far as it's the only reason why Kentucky should continue to exist at all.

I would suggest Shenandoah National Park. Partly for the beauty of the park itself, and partly because the drive from Indiana will go through some gorgeous terrain. I would suggest taking route 33 through Wayne National forest in Ohio. You can stop in Athens as a halfway point, it's a picturesque little college town nestled in the hills.
- If you're going by Athens anyways, you should definitely check out Hocking Hills! It's a very pretty state park with some fantastic waterfalls.
  
  As a Michigander Hocking Hills literally changed my view on Ohio. Incredible place. We did an early spring trip with a cabin and hot tub. We want to go back in the summer during better hiking conditions.
- My wife and I have been to Athens because my brother went to school at OU and I also know southern Ohio because I went to a training school in Chillicothe. We also just drove through northern Ohio to take a trip to Niagara Falls. I don't know if we've had enough Ohio yet or not.
  
  Fair enough, if you end up going that way anyhow, it might be worth taking route 7 along the river. It's minimal Ohio and a very pretty drive.

Note on Yellowstone, Grand Teton is just south of it, and is much smaller, but significantly more dramatic (If you expect to summit anything there without a ski lift, good luck).
As for the east coast, New River Gorge WV is very active with guided activities, last I checked. One of the good places for rafting, also the Greenbank National Radio Observatory is within a days drive. If the Virginia(s) is your direction, in addition to the national parks, you also have the estates of some of americas founding fathers in the area, along with old battlefields that are open to the public, if history is on the menu. I would argue that the best park for hiking on the US east coast is White Mtn. State Forest in NH. Middle of ski country for the winter and summer is peak hiking for the best section of the AT (the Presidentials will kick you and your cars ass, Mt. Washington is up there and you can drive to the weather facility at the top)

I grew up going to the Great Smoky Mountains fairly regularly as a kid, my favorite was always to go to Chattanooga. It’s like a mid sized town in the middle of the Smokies, so it’s super easy to drive out into very rural natural areas during the day and drive back to the amenities of town at night. There’s also a lot of fascinating Civil War history in the area if anyone is a history buff.
The aquarium is great, and I’m a sucker for the incline railway and Rock City (featured prominently in American Gods). Rock City is technically in Georgia, but it’s a 15 minute drive away from Chattanooga so it might as well be right there.
If any of the kids are 8 or under they’d probably enjoy the Creative Discovery Museum. It’s a pity caves aren’t an option, because Ruby Falls is an underground waterfall in the area and very beautiful. The other thing we always did was either go horse riding or zip lining.
Oh, and if you see people selling fruit or veg on the side of the road with hand written signs, stop and buy it. It will be some of the best you’ve had. Same rule for boiled peanuts and apple butter.
- The aquarium is definitely a big selling point. Whenever we go anywhere, my daughter asks if there's an aquarium.
- And it's also not far from Pigeon Forge which is an absolute wild place that also has Dollywood in it (one of the most beautiful theme parks with tons of rides for kids)

Isle royal isn’t very easy to visit. But the parks department operates 2 “national lakeshores” in MI that everyone should see. Sleeping bear dunes is west of traverse city and is super accessible and close to lots of lodging in TC. Pictured rocks is perhaps my favorite place on earth. But, driving up to the UP is a bit of a trek.
- Just looked it up. From us to Copper Harbor (the farthest port) is less than 11 hours by car and I wouldn't really count the ferry in the travel time because that would be part of the experience, so honestly that doesn't sound too bad. Also, we'd drive through both Chicago and Milwaukee if we wanted something to do on the way there or back. It's not a terrible idea, although @Null's suggestion of the Smoky Mountains is appealing since I've driven through them and it was beautiful, but it would also be cool to go somewhere none of us have been. Thanks!
  
  If you are considering either sleeping beat dunes or pictured rocks I would suggest taking the MI route instead of Wisconsin. Less traffic through Chicago and Milwaukee. The drive up the coast on US131 or US31 follows Lake Michigan the whole way and has beautiful scenery.

The boundary waters area is really great, but maybe not for a family. Honestly, I don't think any of these areas are going to compare to Yellowstone in terms of the purely sublime combined with ease of access with children.
New River Gorge is nice, but, like a lot of these options, it's "take your kids for a weekend" kind of nice, not "my mom is paying for a multi-generational meeting at a national park visit" kind of nice.
It's all what you make if it though. Were I you, I would figure out how to go to Yellowstone. Especially considering it's the preference of the person, you know, paying for it.
- I agree.
  OP, just bite the bullet and go to Yellowstone. I live in Illinois and have been to several of these other parks / areas. Yellowstone is on a completely different level and it's not even close. It is one of the best, if not the best, parks in the world. This is not an exaggeration.

I've been to all of those except congaree and if I had to go back to one I'd do isle Royale. Great for introductory backpacking with it's low elevation change and pit toilets at all camping areas. More modern toilets at either end of the island. You also can kayak to one of the the surrounding islands and camp there. If that's not your thing there's a hotel, restaurant, and boat tours of all the shipwrecks. I'm a big backpacker though so if I went back it'd be kayaking and camping on the chain if Islands around one end of the island. Lots of more family friendly options near rock harbor too. Like others have said you can visit lots of things in the way up to break up the trip. I'd do a day at pictured rocks, tahquamenon falls, kitch-iti-kipi, or even a couple nights on mackinac Island on the way up.
- If you get the chance, Congaree is pretty cool if you're ever in the neighborhood in July the fireflys glow in sync!
  Otherwise, if you're in Columbia, just go to the zoo and botanical gardens instead.

Gateway Arch? I think you meant to say City Museum.
- City Museum is awesome, but we've been there twice.
  
  Yeah it is! And in that case, my recommendation is Shenandoah. Tube down to Harper's Ferry and climb Maryland heights. Find one of those trails in the hills that follows a stream that slides down long smooth rocks into cool, clear pools.

I think the Smokeys are the #1 most visited park
- Specifically, Gatlinburg/Pigeon Forge.
  A bit touristy though if that is what OP is going for… but also no real need to go in town either and can just enjoy the park on its own.
  Renting a chalet for a weekend also makes for a good time, too.
- It's the most visited because everyone who needs to drive through that area is visiting it.

Shenandoah is really nice. None of the East Coast parks are going to be as mind-blowing as the famous ones out west. And you can also go to the Great Smokey Mountains area if you plan the route correctly.
- Yeah, that's the real shame. You can't really compare with something like Yosemite. But it's just too far away. Great Smokey Mountains is sounding like the best bet so far. There's stuff in Gatlinburg I know we would like to see as well. My daughter loves any aquarium anywhere and it has one, and there's a pinball museum. My daughter and I both love pinball. I can tell it's a tourist trap, but between that and the national park, I think we'll have a lot to do.
  
  There’s stuff in Gatlinburg I know we would like to see as well. My daughter loves any aquarium anywhere and it has one
  Depending on how much driving around you want to do, it might be worth heading down to the Tennessee Aquarium in Chattanooga. (Take this route for a scenic drive including the Tail of the Dragon and the Cherohala Skyway.) You're also kinda within striking distance of the Georgia Aquarium (the largest aquarium in the US and the 4th largest in the world), although the drive isn't nearly as scenic.
  
  I have lived near Smoky Mountains for years. Especially if you have kids, they will not go bored and the park is nice and not infested with people. (You just gotta go through the people to get to the park). Make sure to get to Clingman’s Dome for nice pics. There’s lots of trails for all experience levels too.
  
  Ripley's aquarium is in Gatlinburg, we went there last year during our vacation to the Smokey Mountains and thought it was awesome. We haven't been to a lot of aquariums, but it's the best I've been to.

My partner and I have a goal to visit all the national parks, so we've done almost all of these recently.
Mammoth cave is by far my favorite, but I'm a computer nerd, and a giant hole in the ground is my natural environment, so there may be some bias.
Gateway arch is okay, but not worth a long drive in my opinion.
Got springs was very neat. It has a different feel than the other parks I've been to. It is a bit "touristy".
New River Gorge is amazing. Be aware that any hiking there is going to take a lot of vertical travel, a 1 mile trail there is more like a 3 mile trail in a flat area. That being said, it has a lot of wildlife, picturesque views, unique structures (big ass bridge) a cozy little town (Fayetteville reminds me a lot of downtown Bloomington) and plenty of activities (hiking, rafting, zipline, tree top obstacle course, history, shopping). Easily a top recommendation.
Congaree. Probably the most wildlife I've seen at a park. Snakes, salamanders, pigs, owls, fish, skinks and a shit ton of fuzzy caterpillars. Very buggy, but honestly the mosquitoes weren't as bad as everyone said they would be. Also highly recommend. The park has a long boardwalk as it's "main" attraction, which is very accessible. Saw lots of wheelchairs while I was there. I'd recommend taking a kayak tour as well.
Great Smoky Mountains is a great choice as well. Has that authentic "National park" feel. Very big (for an East of the Miss. park) Gatlinburg is nearby for lodging and other activities. You could easily spend a week here between the two. I like more unique "unique experiences" but GSM would be an awesome choice if your family isn't very adventurous.
Don't forget some state parks while driving through. Red River Gorge in KY is one of my favs, and easily worth the trip on its own. You've got Natural Bridge State Park right there, and the skylift is the best way to get to the top. Shout out to Daniel Boone coffee shop, best breakfast burrito I've ever had, but the line goes out the door, so be prepared to wait a little. FYI, this area had a bad fire last year, not sure about the extent of the damage.
All in, I'd say do GSM. Solid pick, easy to get lodging, lots to do if the weather doesn't cooperate. On your way over, hit up Red River Gorge on the way if it's not too far out.
- Thanks for all of that! I appreciate it! I think based on your and everyone's recommendations, I'm going to suggest the Great Smokey Mountains.
  My wife and I really wanted to do Mammoth Cave either last year or the year before, but my daughter said absolutely not. We've offered since to take her to closer-by Marengo Cave or Blue Springs Caverns, both of which are also terrific, but she says she is never going into a cave as long as she lives. She didn't even want to go into the tunnel at the Niagara Falls Power Station when we went to the Falls earlier this year and that wasn't even a real cave. And, of course, she hates it every time we drive somewhere and have to go through a car tunnel.
  It's weird because she's not otherwise claustrophobic. She just doesn't like the idea of rock over her head.
  
  GSM is a great choice! There are a few mountain-side tunnels you have to drive through in the park, very short, maybe 50 yards at most. Hope your daughter is okay with that.
  Shame about the caves. I know it probably won't help comfort her, but more people are going to die in car wrecks in Indiana today than have ever died from cave-ins at Mammoth cave.
  Clingman's dome at GSM is (or was) the highest point in the Appalachian mountains. Def visit in the evening if you want amazing views. Cades Cove usually has bears and is a driveable trail. We saw 2 bears and a bear cub, plus some deer and Elk on the drive.
  I saw in another post that your fam like aquariums, there is the Ripley's aquarium in Gatlingburg, which was a little expensive IMHO, but it is a decent aquarium.
  Skybridge is a ton of fun, but very touristy.
  There is a little GSM trail at the edge of Gatlinburg that my wife and I walked at night. We frequently do this in parks. Bring red lights to preserve your night vision, and a black light because there are tons of centipedes that glow a ghostly blue when hit with a blacklight. We also blinded the shit out of a few mice. Try not to do that....
  Also, depending on the time of year, do some road cruising in the morning or early evening. You'll see lots of wild turkeys, deer, and snakes. Please be careful not to run over the snakes.

Niagara Falls isn't too far out of your area, 9 hours for the American side, maybe 10 for Canadian. Walking on the pier under the Canadian side is incredible, if you are willing to pay for the ferry ride.
- We actually were just there in August. It was amazing!
  
  I was really disappointed that I didn't get to go a second time, during a maintenance a few years ago that had the entire Canadian side blocked dry. Would have been so cool to see how the rock has been shaped.

I live in VA and I find Shenandoah to be the most boring national park I've visited. Unless you can catch it during foliage season, I'd go somewhere else. That said, Skyline Drive is a mess during foliage season. Great Smoky Mountains is nice.

I've done quite a bit of camping in the UP of Michigan and have been really impressed by how much beauty can be found up there. Doing Pictured Rocks, Tahquamenon Falls, and Kitch-iti-kipi on a 4 day weekend trip is pretty incredible. So I'd do Isle Royale and hot up a few more landmarks while being up there. Sounds like a good clean inexpensive trip!

Having been to all but voyageurs go to either the smokey mountains or new river gorge. I was going to put mammoth cave on that list but, you know fears and stuff. I will say mammoth cave does not feel like your typical cave, it’s way larger, and has been adapted for tourists.
- I've been to Mammoth Cave before myself and really loved it, but she doesn't care how big the cave is. She says she just doesn't like the idea of all of that rock above her head. We tried to convince her a couple of years ago, but she's dead set against it. Smokey Mountains sounds like the best choice.

based on name: isle Royale

Not national parks, but Elephant Rock and Johnson Shut-Ins in Missouri are both pretty neat.

Not a national park and kind of on the outskirts of your radius, but any thoughts on the Finger Lakes region of NY? Lots of hiking, waterfalls, lakes, etc.
Ithaca
Watkins Glen

I've only been to Indiana Dunes and Voyageurs, but they're both nice enough. Voyageurs is like, bring a canoe and camping supplies, though, not necessary a day trip.
Consider some state parks! Turkey Run in Indiana is gorgeous, and Pipestone State Park in Minnesota is beautiful, historical, and unique.
- We're pretty close to Turkey Run and we always tell people to please go to Turkey Run and do not go to that horrible Shades State Park which is definitely not nicer in pretty much every way imaginable and will have an unpleasant lack of a giant line of people going down the trails.
  
  Haha fair. I really liked Turkey Run when I went there, is all.
  But seriously, check out Pipestone.

Isle royale is on my list because its so wild. May not be great for a family though because its so hard to get to.

Hot Springs in Arkansas.
I have no idea why the St. louis Arch is listed as a national park here. More like a national arc at best.

Personally I'm partial to looping them all into a trans continental high speed rail network, make them all visitable in simple order