I'm aware that a lot of modern speculation about pirates is false, but were the parrots real? It seems a little far fetched that pirates owned parrots, so it might be real. Don't question my logic.
I'm no expert, just the last few years I've been reading pirate histories.
The Pirates' Who's Who: Giving Particulars of the Lives and Deaths of the Pirates and Buccaneers, Philip Gosse, a fairly good read, encyclopedia of all notable pirates, does not describe any parrot as pet or sidekick, on board ship or on land.