The episode "Not-So-Great Great-Great-Grampa". At first, it seems like the Grandfather Paradox moves like they normally do. But think about this: If Phil had never been born, he never would have travelled to the past and given Kyle the idea for the prank, meaning Kyle would have stayed at H. G. Wells, never gone to military school, and never met Petunia. Which means that Phil would have never been born.What.
There's a bit at the end where they mention that Petunia likes Kyle, creating a Stable time loop
I get that. The problem is that there's no way for the paradox to be possible in the first place. Either Phil doesn't get born, or... he doesn't get born. I'm okay with Timey Wimey Balls, but this makes no sense.
No, see, there was no zero-th iteration. Phil gets born, and causes his own birth in the process of his existence. There is no "doesn't get born". It's like a My Own Grandpa situation, where everything had to be in place. The series isn't terribly consistent across the series as to whether they change the past or already went back to our present (unless the expectation that they can undo their trip to the past, as in the time capsule episode, is their own misunderstanding), but this is a perfectly logical series of events within the episode, from what I recall. Unless that's the episode where Phil fades like Marty McFly, in which case disregard everything I've said relative to the subject at hand.
Just because it was that prank that got him into military school this time doesn't mean that he wouldn't have done another equally dangerous prank that got him there. So it's quite possible that if Phil didn't come, he would've just performed other pranks until he finally got sent to military school.
There's an episode where they sent a psychotic robot to the 21st century. Why would they send a psycho back in time purposely? What if she changed something?