....Paarthurnax after the Main Quest is completed? I didn't want to kill him when I was charged to by Delphine, but now I don't care. I'd like to experience more of the Blades. I went to the Throat of the World and attacked him, but he just sat there on the word wall. His health went to zero, but he didnt die. Is it a bug, or did I just lose the chance?

I'm not 100% on this... So check out Google and such...

Use dragonrend on him and kill him once he lands.

Would Dragonrend put him on the ground even though he is already sitting on the word wall? I brought him to zero health and he just sat there looking at me with his sad dragon eyes. Perhaps because he is sitting on the wall makes him remain unkillable....

I think I remember reading that he becomes essential after the MQ. Not 100% on that though.

The Elder Scrolls Wiki mentions using this as well.

Hm. Well I certainly don't think you "missed the chance" (I don't think this can happen). But when I decided to kill him, I had a conversation with him (there's dialogue where you tell him the Blades want him dead, and he says they're right to not trust him) but afterwards he continued to just sit there on his little dragon couch wall thing. I fired one arrow into him, health dropped, he doesn't react. Two arrows, same thing. On the third arrow he finally became aggro and started flying. Dragonrended and he died like any other dragon.

Perhaps it's a little glitchy.

Yes! That's exactly what happened except he didn't do a thing when his health got low. I had the conversation with him about the Blades wanting him dead, as well. I'm going to reload the save and Dragonrend him. Hopefully something will snap.

I found that attacking him after the MQ wa over had no effect. He never took off, just sat there and took it until his health went down to zero, and still didn't die. I didn't think to try Dragonrend on him because he never flew. I tried multiple reloads, but always got that result. When I tried it as soon as you got the order to kill him, the first attack went the same way. I reloaded and tried, and this time he took off and Dragonrend etc worked, so I killed him. There seems to be a bug, and if he doesn't take off, he's not killable.

Conclusion: I went back to Paarthurnax and shouted him with Dragonrend. He didnt do anything so I shot him and then shot him again. With the second shot he got annoyed and attacked me. Paarthurnax = dead.

I've read if you use the storm call shout, he eventually comes down and is killable. That's how I killed him, but wasn't after the MQ was over.