No the Pirate parrot is not a lie... well not completely a lie. There is evidence in manifest of a Parrot trade so it is not within the realm of fiction that Pirates would HAVE parrots and trade them. But they probably did not take them in as pets and have them squawking about on ships and shoulders. Having a pet would not be easy for a pirate either. Their lifestyle of leaving port for months at a time, gaining their loot and then resting a few months in a different port until they ran out of money made them a bit to mobile. So Pirates and and parrots go together but not as pets.