Honestly, if you're looking for the absolute freshest 'South Park' fanfic, your main hubs are gonna be Archive of Our Own (AO3) and FanFiction.net, but they serve different purposes. AO3 has this incredible tagging system that lets you filter by the exact pairing or trope you want, and it usually gets the new stuff from bigger event weeks like new episodes dropping or holidays pretty fast. FanFiction.net still has a dedicated, if slightly smaller, 'South Park' base, and some writers cross-post, but their sorting is way clunkier.
I'd say the real-time updates often happen in smaller, less obvious spaces, though. Tumblr tags for 'south park fanfiction' or '#southparkfic' are a firehose of WIP snippets, ficlets, and links the minute someone gets an idea. Writers will drop a 500-word drabble there based on that night's episode before it's even fully edited. And don't sleep on Discord servers dedicated to specific ships like Creek or Style. Those channels are LIVE. People will literally post paragraphs as they write them in a dedicated fic channel, asking for immediate feedback. That's as updated as it gets—raw, unbeta'd, and straight from the brain to the screen.
My routine is to check AO3 for polished, complete works, but I lurk on Tumblr and a couple Discords for that immediate, buzzing 'just saw the episode and had to write this' energy. Sometimes the best, most unhinged concepts start there before getting a proper AO3 home a week later.