There is a little game called Borderlands where every questgiver is a smug oblivious parrot giving you instructions casually as you fight through legions of headshot eating enemies that spawn behind you.

In Risen 2 you can use a parrot in battle to distract your enemy.

There's a parrot in Treasure Adventure Game.

In Treasure Adventure Game you have a parrot sidekick.

In Nancy Drew Curse of Blackmoor Manor there is a parrot called Loulou, whom you need to talk with to solve a puzzle.

If I remember correctly, Nancy Drew Ransom of the Seven Ships also has a parrot. I don't remember if that time the parrot is important for the plot, or just a random "npc" though.

As far as I remember yes.

In Captain Morgane and the Golden Turtle, you get a parrot in the latter part of the game in order to reach the Golden Turtle

I learned about Plato from the parrot in Fate of Atlantis, possibly the most well-read parrot in a game.

But my number one favorite game parrot is Feathered Fleabag from Runaway 2 Dream of the Turtle. He wears a pirate hat and an eye patch and he wants to be fed seeds or else he won't tell you anything. Great voice acting, very convincing.

Serious Sam II had parrots as an explosive weapon.

THAT COUNTS!

The Wii version of Sid Meier's Pirates has a parrot in two player mode. The second player would control the parrot during the sword fight sequence between the ship captains and you can actually attack/distract the captain which would give your friend an opening to attack him.

Jazz Jackrabbit 1 - the sidekick is a parrot (you must free him from cage). Also, you can be a parrot. And in Jazz Jackrabbit 2, the parrot sidekick also can be found, although rarier.

Don't forget Treasure Adventure World. It's still under development (demo is already available), but it counts.

hmmm you can get an pet parrot in terraria XD