Hmmm... did you get Odahviing's name from Esbern? That's when he told me to do it. That was on a playthrough where I had finished the CW before I got that far in the MQ, so I never did Season Unending. On the one where I did Season Unending, he gave me Odahviing's name during one of his speeches at High Hrothgar but I deliberately avoided engaging in any other conversation with him or Delphine so I'm pretty sure I never got the "orders" about P. I never went back to SHT on that character so there was never another chance for it.
At any rate I believe you have to speak to him (or maybe Delphine) after you get to the point where you are trying to figure out how to get a dragon to Dragonsreach. If you are doing Season Unending, you've reached that point already anyway. But you don't have to do Season Unending, just talk to him about trying to capture a dragon. Maybe you have to pursue the convo further in order to get to the part about P.?
I keep thinking there's another way to get Odahviing's name but if so I haven't had it come up in dialogue options with anyone else yet.
Edit, from UESP: http://www.uesp.net/wiki/Skyrim:Paarthurnax_(quest)
This quest is given to you if during http://www.uesp.net/wiki/Skyrim:The_Fallen, you talk to Delphine or Esbern; it will start automatically at the end of the http://www.uesp.net/wiki/Skyrim:Season_Unending anyway, if one is held and you haven't yet spoken to him.
Makes it sound like if you don't get it during that time frame, you don't get it at all. Can that be right? Could you just continue on being "friends" with everybody after the MQ ends, because you never even got the quest to kill P.? Surely if you go back to see the Blades later, they would bring it up, no?