Ah, this is a great question. One of the main comparisons between Aristophanes and South Park will be the highly political nature of their respective forms of comedy. Aristophanes satirized the politics of Athens, while South Park's creators make lots of fun of American politics.
Aristophanes and South Park's creators blend reality and the supernatural. In Aristophanes' Frogs, Dionysus travels to the underworld and judges a poetry contest between Euripides and Aeschylus. In South Park, Jesus and Satan have a boxing match.
In both Aristophanes and South Park, the heroes frequently go on magical journies. In Aristophanes' Peace, the hero flies to heaven on a giant dung beetle to bring Peace back down to earth. In South Park, the boys fly to various places (e.g., Afghanistan, Scotland) and have wild adventures.
Aristophanes and the creators of South Park also rely on scatalogical and sexual humor to delight their audiences. Jokes about "poop," "pee," and the human genitalia have been making people laugh for 2500 years.
Plays like Aristophanes' Lysistrata, in which women demonstrate the power that they hold over men, reminds me of the South Park episode when Bebe goes through puberty, which causes all the boys in the school to start behaving like animals.