Cartman is the joke because his character is an asshole bigot, that usually faces repercussions. We laugh at the absurdity and when he finally gets what's coming to him. He is the shitposter, the troll, that asshole kid we went to school with that quoted unfunny Opie and Anthony bits with Jim Norton during lunch. There's a cathartic feeling seeing him be the butt of the joke without actually realizing it.
What's the joke here with Donda West, though?
All I see is them making fun of a black man showing love and appreciation for a mother that loved and supported him, who died tragically, using a medium that he hasn't had any real involvement in to pay artistic tribute to her. Which he then wished to share with the world.
It isn't about thin skin. It's about the 'joke' being a mean-spirited jab at someone. You know how we can can tell its mean spirited? Because there is so much fucking more material to pull from in order to make fun of Kanye. His shoes, the near-disastrous launch of Pablo, his horrible clothing line, his marriage to a Kardashian, Taylor Swift, his outbursts, the list keeps going...
But what is the best that Trey Parker and Matt Stone can come up with? Well, Kanye West was working on a video game to show the world how he pictures his mother, an angel ascending to heaven. Oh and he said nobody wanted to make that game with him and he had a hard time finding a partner. Seeing as the game hasn't come out, I guess he wasn't too far off.
So they make fun of that and her. A mother that wasn't a celebrity but an english teacher, with racial imagery and the fucking mammy voice.
What a great fucking joke.