In Ratchet & Clank: Tools of Destruction and Ratchet & Clank: Quest for Booty there's a NPC called Smuggler with a speaking parrot on his shoulder.

"Bawk! Kill them now sell their kidneys! Bawk!"

http://ratchet.wikia.com/wiki/Parrot

Spellforce: Demons of the Past has a parrot that you have to find for a ghost pirate, who later gives you a new quest for... something.

In Magicka Wizard Wars, you can get a staff that has a Parrot on top of it.
This Parrot will squawk silly things whilst you kill other players. Some of said squawking, is barely disguised insults.
Makes me cackle every time as I play :D

http://steamcommunity.com/sharedfiles/filedetails/?id=520755599

I might be wrong because I am basing this on a distant memory, but in Diablo 2, the jungle act, act three I think, there are parrots flying around the jungle...lol a stretch, but a possible sighting of multiple parrots.

I know you wanted parrots that are a bigger part of the story or have some influence in the story, but this is all I could come up with. Most of the posters already posted about my parrot memories. :oP

And I think there's a parrot somewhere in Rayman 3: The Great Escape, as the first few levels are pirate-themed and I think I remeber one being there somewhere, perhaps on the shoulder of the big boss of the mechanical pirates' flying ship area.

IIRC, there was a parrot in Lost in Time.

wouldn't it be awesome, if the devs of hatoful boyfriend just release a dlc called parrots?