That's it? You just need to say "you know this actor character we call Tom Cruise in our show, and does the same stuff as the real actor Tom cruise in real life, it's actually a fictional character unrelated to the real Tom Cruise in real life" and you Gucci?
Keep in mind that a celebrity can send you and me, regular joes, a C&D and we'd likely comply simply because we lack the resources to sustain a challenge in court.
What stops celebrities and organizations from suing South Park creators is likely the opposite: they have money and a legal team.
The same thing happened with John Oliver when he talked shit about some coal mine owner that was notorious for suing people. The mine owner served them from a court that had friendly laws but they were ready. And they had insurance to pay for it.
This is really only to prevent them from suing and claiming that the show is representing the actions of celebrities on the show as true fact. If I publish a newspaper article saying "Donald Trump strangled a baby" without evidence then that could be actionable, as it's a factual claim. But if I showed Donald Trump strangling babies as part of an obvious parody, then that would be protected speech. So their disclaimer is basically just to make it obvious that it's a parody. It's not a requirement, but they want to cover their asses.
If it's all stuff he really does in real life, how could he sue over it? If it's factual, you can't sue. Well you could but you'd lose because South Parks lawyers would say "show me what we said that wasn't true"
If it's its fictional and they clearly state at the beginning of the episode that it's fictional and you should not assume it's true, it's going to be really hard to convince a judge that they were trying to trick people into thinking something was true in order to harm you.
The first amendment allows you to make fun of people, especially if you say "we're just being silly and none of this is serious" beforehand