Q:

What did Paarthurnax do when Alduin returned?

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?

A:

Paarthurnax didn't do anything directly when Alduin returned; he waiting at the Throat of the World for the Dragonborn.
Paarthurnax had always been subordinate to Alduin; he likely knew a direct confrontation would not go well for him. In fact, he seems to know full well that the only way to defeat Alduin is for the Dragonborn to learn the Dragonrend shout, which no dragon can actually use.
So, his plan was to be there to help the Dragonborn, so the Dragonborn could kill Alduin. He picked that spot because of the metaphysical significance it had: it's the place where rhe Dragonborn could use an Elder Scroll to learn Dragonrend. It's also a place where the Dragonborn was likely to come, to learn shouts, but unlikely for anyone else to stumble upon by accident.

