Quite Strict
The TSA will need to screen your bird and ask questions. Part of the process will be for your bird to go through the Body Scatter and check its id.
Then the flight attendant will ask you if your parrots can eat their nuts. LOL
The serious answer would be to call 1800-435-9792 and ask if your bird can travel with you.
Enjoy your flight. BTW I love the photo you posted of your bird very pretty.