The Emerald Parrot is not aggressive, and will only attack if you attack it first. It charges toward you in a straight line and can do a considerable amount of damage to new players.
The best time for a Warrior to attack is right after it charges.
For ranged attacks, just keep shooting and roll when it charges toward you.
You can also use any of the Shout Skills such as the following to make them fall to the ground so you can attack. This is particular useful if you are a warrior.
You can also use any of the throwing knives/axes to attack from range.