South Park usually makes fun of both sides (at least back then). A good example of a South Park episode where they made fun of both sides and even sided with both of them for particular reasons would be "Naughty Ninjas" where they made fun of both sides regarding police brutality, where they made fun of police incompetence as well as people's dumb reactions to said police incompetence. Right now, modern South Park still makes fun of pussy liberals (e.g. the aforementioned trans athlete episode), but there are times where they cater to pussy liberals, not in terms of "they made fun of both sides and sided with pussy liberals for this reason and also sided with conservatives for this reason" like what they always did, but rather completely catering to pussy liberals sometimes.
The most notable example for some would be the South Park creators decide to reverse their decision about Al Gore and made fun of Americans for putting the blame on China for global warming, but for me the "Splatty Tomato" episode (which for me, was when South Park jumped the shark and the main reason isn't because of political leanings but rather, the abysmal writing and unfunny repetitive jokes etc.) was also a notable example. (Blaming Trump's victory on "The Whites" and nobody cares about "The Whites"... how subtle.)
It's shit like this is also why you get shitty articles like these:
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The funny thing is, the trans episode was about men exploiting a loophole to basically force women out of their own sports. You could have interpreted the whole thing as an attack on Toxic Masculinity and the Patriarchy taking over everything, but no, they went for the "Transphobia!" angle, when that's really not what it was about.