Paarthurnax is the leader of the greybeards who lives on top of the Throat of the World. He tells the Dragonborn he specifically stays there because it is where Alduin was defeated in the First Age, and where he would return whenever he got free from drifting through time
Some hoped he [Alduin] would be gone forever, forever lost. I knew better. Time flows ever onward. One day he would surface. Which is why I have lived here. For thousands of mortal years I have waited. I knew where he would emerge, but not when.
As we know, Alduin did return, prompting the events of the game. But what did Paarthurnax do when Alduin returned? He never mentions confronting him when he returned. He doesn't seem injured from fighting him (although I'm guessing if Alduin had beaten him he would have killed him, rather than leaving him as a mentor for the Dragonborn). At first I thought Paarthurnax was going to betray the Dragonborn, and was waiting for Alduin to return so he could help him, but that's not the case.
So what happened when Alduin returned? Why did Paarthurnax wait at the spot he would return to, and what was his plan for when it happened?