I think the argument was that if your parents are not naturalized citizens, then that means they're not 'subject to the jurisdiction thereof'. So like, if the parents 'owe allegiance' to their previous country then it makes the kid a citizen of the parents' original country, and not the U.S.
But it's all actually bullshit to try and justify not wanting brown people to be U.S. Citizens tbh.
I mean, I'm a millennial in mid-thirties and I'd never heard of anyone actually doing a party like this either, so even in the 90s/00s these were things that seemed like 'stuff they used to do before'