The Treasure Island character is pretty much "to blame" as the mold of many later stereotypes about pirates. The eye-missing wood-legged parrot-affixed sea dog that is long John Silver has struck the imagination of many. One vivid and colourful portrait, however, doesn't make up for a good historical source. It rather becomes a cliché and a cliché looses its evocative power or must be renewed, such is the case in Pirates of the Carribean, where the Long John Silver cliché is parodied in Cotton's character but also refreshed in Barbossa. The latter traded the parrot for a monkey, another exotic animal that was often found in the Carribean and sold at good prize in Europe. Barbossa also lost his leg later in the franchise whereas Cotton is seen closing an eye while piloting the Black Pearl. Both of them are obvious avatars of Long John Silver.
Pirates followed trade routes. They mostly attacked ships to get their hand on precious cargo such as sugar, tabacco, leather, spices and slaves. They could sometimes hit a treasure gallion but that was rare for they were well protected. Those constant attacks, however, really weakened Spain (their favorite target) on the long run.
The Carribean often makes us forget that pirates also operated along the shores of Brazil and West Africa, but also in the Indian and the Pacific Ocean. As such, pirates often coasted tropical forests which are the most favored natural habitat of many parrots, but the bright and colorful parrots that we see in movies would mostly be found in the region of Veracruz (Mexico) or in Brazil. Grey parrots, which are far more gifted to imitate human speech but never made to a major motion picture (to the best of my knowledge), were found in Africa. Other kinds of parrots could be found in various parts of the world but they too wouldn't fit the hollywoodian cliché.
Nevertheless, exotic animals weren't a most prized catch. They couldn't be put to work like slaves. They required special care and feeding. They could be sold for a very good amount but getting ahold of sugar or tabacco demanded far less logistical headaches and offered more promising rewards.
Pirates ships often sunk and their crew regularly fell ill or starved. Records are full of stories about the fear or the experience of starvation. Asking for help from Natives wasn't always a possibility either. Any pirate crew that'd have transported exotic animals would first have had to make sure that they'd have enough food on board. Sometimes they were desperate enough to demand food instead of seizing the usual leather, spice and sugar. Dogs are barely if never mentioned aboard ships. Cats seldom found their place on board as rat hunters. Either way they'd probably been eaten by the crew if the occasion arised. So would have been any other exotic animals if necessary.
Nevertheless, exotic animals did come ashore in Europe from the Caribbean, South America, Africa, Asia and Australia at a time when pirates roamed the sea. Exotic bird markets were very popular until they were banned in several countries in the 20th century. Between the 16th and the 19th century, they entered the imaginary landscape of treasure islands, exotic landscapes and eye-patched wood-legged adventurers. Did pirates often sailed with parrots on their shoulders? Probably not. Much has been written and fantasied about them and very little is true. However, we can't rule out that there was one pirate out there, between 1550 and 1750, who ventured the sea with his beloved, talkative and colourful bird.