When you first meet Paarthurnax he tells you that he had stayed at the top of the mountain waiting for Alduin to return. Well, Alduin showed up at Helgen obviously flying down from the top of the mountain. Did Alduin and Paarthurnax have a brawl before that? Cause it's never mentioned. Or am I just forgetting something?

Maybe, but Alduin would win even if they did. But if they did fight, I can't understand why Alduin wouldn't kill Parth, because he is obviously the stronger of the two, and Parth did screw him over by teaching thee Thu'um to mortals. So maybe they didn't. We don't really know.

Paarthurnax was out shopping for some groceries and he missed it.

Maybe Alduin reappeared somewhere else. The time wound is a special place where it's possible to catch a glimpse of the past, like a crater left by an explosion. But in the dialogue with Paathurnax it wasn't mentioned that Alduin would reappear there. Only that he was bound to return in the place.

Come to think of it... The "Throat of the World" is the highest mountain so the perfect seat for someone who thinks to be the ultimate ruler of the world,

A fight between dragons is a conversation. Maybe they did "talk" when Alduin showed up from the past. Then Alduin flew off in a rage and arrived at Helgen, which is just "coincidentally" where the dragonborn happens to be. Paarthurnax has been watching the world from his perch so HE is probably the one who told Alduin (who only just arrived) where the newest dragonborn was to be found. Paarthurnax does seem to know in advance why you show up at Throat of the World and he is not talking to the greybeards..