Cranking these out in my free time...

But here's one special guy:

Groucho Parrot:
Hoping to complete Chico and Harpo in the near future!

On with the flag parrots!

Checkered Flag:
Romania:
Luxembourg:
Monaco:
Morocco:
Algeria:
Libya:
Egypt:
Brazil:
Iraq:
Iran:
Israel:
Palestine:
Jordan:
Lebanon:
Syria:
Saudi Arabia:
Greenland:
Cuba:
Guatemala:
Belize:
El Salvador:
Honduras:
Nicaragua:
Costa Rica:
Panama:
Jamaica:
Haiti:
Dominican Republic:
Puerto Rico:

Some remakes:

United Kingdom: (I made it from scratch like the others, but this is very similar to the pre-existing UK parrot)
LGBT Pride: (flag pattern moves along with parrot)
Transgender Pride: (ditto)