Paarthurnax (quest)

Walkthrough
This quest is given to you by the Blades. They found out that a long time ago that Paarthurnax was once an ally of Alduin.

Kill Paarthurnax
They ask you to kill him to make sure that he will not betray Skyrim.

You can complete the whole main quest and then defeat him thereafter. He is not a difficult battle, just simply bring him to the ground (via Dragonrend) and attack normally and he should be defeated with ease.

It would probably be best to kill Paarthurnax once you have meditated on all the three words possible and have all shouts unlocked, as the Greybeards will not talk to you (and in turn not provide you with word wall locations) after you kill him.

Talk to Delphine
After killing Paarthurnax return to Delphine, she will be delighted that the Dragonborn did the right thing, at least in her view.