In the United States, parody is protected by the First Amendment as a form of expression.

Parody is protected under US law.

People can (and do) sue, but they lose every time because it's easy for their well resourced corporate legal team to prove the show is (obviously) parody and thus, protected free speech.

Simple answer is they are careful about what they say and have good lawyers that review it.

A few examples.

Calling Tom Cruise a fudge packer in the context of him being in a bathhouse could eaisly open them up to liability for calling him gay. But doing it in a fudge factory while showing him putting fudge in a box gives them a clear defense that they meant it literally.

Simmaraly telling him to come out of the closet while he is actually in a closet provides cover.

Making things so absurd that a reasonable person wouldn't believe it and know it's a joke also works. So having Barbara Streisand aquire an artifact that makes her into a giant robot monster works but something plausible wouldn't.

Having Kanye open up and admit he is a gay fish is absurd enough to provide protection. However they probably couldn't get away with him simply coming out as gay.

Of course the genius of south park is they use these legal protections in ways that make the story funnier and not just for cover.

Reminds me that Scientology sent "private investigators" after Matt and Trey to find dirt on them, but ended up with nothing

https://www.avclub.com/scientologists-spent-a-long-time-investigating-south-pa-1798228087

They do get sued pretty often, but the law is on their side. The more absurd the circuimstances the easier it is to get away with as Parody, the only time you're in hot water is if the viewers in any sizable metric would be fooled into thinking the things portrayed in the show were true when in reality they weren't. In fact, when South Park revealed everything about Scientology's internal beliefs they were almost sued but Scientology backed down because what South Park said in the episode was actually verifiable truths that could be backed up with evidence in court.

Southpark isn't sued as far as I know, but they have received massive criticism and even death threats from terrorists organizations.

keep in mind, we're talking about the show that toppled Scientology.

just, for the record.... part of how it's able to make fun of shit is because they're usually correct about the stuff they're mocking.

Saying 'Biden is a baby-sacrificing pedophile' is defamation. saying 'trump is a rapist and a fascist' is not.

further, both parody and satire are in fact protected speech. at least, for the moment.

I always find it funny that this got them in trouble

This didn't

But I'm still pissed this didn't get aired:

https://youtu.be/mDmOGN-x0ls?si=yzsO2Fh_d-tc-fP-

It's a very fine line they would have to walk. It must be believable to the average person that the claims are true. It must not actually be true. It must be done with (the appearance of) malice. It must not be done as a criticism/satire of the target and their actions.

And on top of that, their publicist/PR must think a lawsuit will get them more than they lose. Once it's aired, it is out there forever. It could then be the one that everyone seeks out and shares with friends, as "the one that Tom Cruise sued to get rid of". This is known as "The Streisand Effect".

As others have pointed out, US first amendment laws generally protect shows like South Park because it's generally understood that the characters in the show that resemble real people are parodies, and the show runners aren't stating a fact that the real person said or did a thing in reality.

Funnily enough, the UK has much stricter laws about defaming people - the country has a strict class system, and it wouldn't do if poor people could embarrass rich people - there is a significant carve out for "vulgar abuse". If I was to go on TV and (for sake of example) called Boris Johnson three shit-stained jugs of fetted piss wearing a trench coat, that would be ok, because people understand that to be a euphemistic insult, not a literal statement of fact. If I went on TV and said that he was a drunk, that wouldn't be - unless I can prove that he is an alcoholic, he could sue me for libel. The outcome of this is that an equivalent show to South Park could be made in the UK, it would just have to be utterly filthy