Ship parts are an item like med kits, but for ships. You can buy them from some merchants or sometimes loot them from destroyed ships.If it's in the cargo hold of your ship you can use them. You can enter the ship's hold a number of ways. In the start menu go to ship and you see a button to enter the cargo hold on the bottom right. There you can store or take items from your cargo hold like you're trading with a companion, as long as you're in a radius of 250 meters of your ship. There's also a blueish computer screen in your ship's cockpit where you can enter the cargo hold, in the frontier it's on the left on the wall just before your piloting seat.
There’s a type of aid item simply referred to as “ship parts.” Places like Jemison Mercantile and UC Distribution Center sell them sometimes, and you may even earn them by helping UC/Freestar ships beat up hostiles in random encounters.
Edit: and don’t keep them in your on-foot inventory, they weigh 10 kilos each.
Each ship has a cargo hold that you can dump stuff in. From the 2 ships I've been in (Frontier and Shieldbreaker) you can access the cargo hold from a computer to the left of the pilot seat. If you buy or find Ship Parts (the name of the item), you'd have to go to your characters Aid section, and move them to your ship's cargo hold. Then the button shouldn't be greyed out anymore.
I want to add that you can enter the cargo hold by going to the Menu (Tab on PC), then click on lower left where your ship is and see the cargo hold (F on PC)
You can also move things from the cargo hold to your inventory even if you're outside of your ship, I don't remember if you can do so the other way around.