Everywhere I go, I've seen that the way to unlock the second part of this is to beat it in under 5 minutes., which I have done REPEATEDLY and it still hasn't triggered.

There have been a couple of other Parallel Quests where I've met the requirements and it hasn't triggered either, so I guess my main question is actually: is this a glitch or does meeting the requirements for a mission only POSSIBLY trigger it?

I can't speak to this specific mission but when I was farming Broly for his gear, the second fight against him would often not trigger despite doing everything exactly the same as when it would trigger. So, yeah, it's random.

It only possibly triggers it. RNG ontop of RNG. Just keep at it if you care enough.

Pray to RNGeesus

Well allllllllllllllright then.