South Park

It wasn't her looks they were picking on, it was her plastic surgery. Getting plastic surgery is not your looks. And they didn't even make fun of it, they just depicted it. This is what a standard run-of-the-mill caricature looks like, you emphasize features that stand out to a further degree. That is how every single celebrity has appeared on the show, unless they did photo cut-outs. But even their very best real-life friends are done in caricature.

She saw it as making fun of her looks because of how she perceives herself and how her ego handles it. It is apparently a sore spot to her.

Edit: it's actually much more likely she is trying to redirect and discredit. If she pretends that's what the episode was about, it's a less defensive position. So it's actually in effect similar to the reason they make straw man arguments to attack instead of real ones normally. They can't refute 'your' real point, so they fabricate a different one that is much easier to attack and ascribe it to their opponent instead of their actual point.