Parrots did live in the Caribbean:

<img alt="" src="https://lemmy.today/pictrs/image/fa5b413a-09f5-438a-bf7a-be636bd9e142.jpeg">

And the Caribbean was an important center of piracy for some time:

en.wikipedia.org/wiki/Golden_Age_of_Piracy

The Golden Age of Piracy was the period between the 1650s and the 1730s, when maritime piracy was a significant factor in the histories of the North Atlantic and Indian Oceans.[1][2]

Histories of piracy often subdivide the Golden Age of Piracy into three periods:[3]

1. The buccaneering period (approximately 1650 to 1680), characterized by Anglo-French seamen based in Jamaica, Martinique and Tortuga attacking Spanish colonies, and shipping in the Caribbean and eastern Pacific to western Pacific.[3][4]
2. The Pirate Round (1690s), associated with long-distance voyages from various Caribbean and North American ports to established bases in countries like Madagascar, in order to rob Muslim and East India Company targets in the Indian Ocean and Red Sea.[3][5]
3. The post-Spanish Succession period (1715 to 1730), when English sailors and privateers left unemployed by the end of the War of the Spanish Succession turned en masse to piracy in the Caribbean, the Indian Ocean, the North American eastern seaboard, and the West African coast.[3][6]

Narrower definitions of the Golden Age sometimes exclude the first or second periods, but most include at least some portion of the third. The modern conception of pirates as depicted in popular culture is derived largely, although not always accurately, from the Golden Age of Piracy.

searches a bit

It sounds like much of the image derives from the fact that Long John Silver, a fictional pirate, had a parrot.

en.wikipedia.org/wiki/Long_John_Silver

Long John Silver is a fictional character and the main antagonist in the 1883 novel Treasure Island by Robert Louis Stevenson. The most colourful and complex character in the book, he continues to appear in popular culture. His missing leg and parrot, in particular, have greatly contributed to the image of the pirate in popular culture.

Long John Silver is a cunning and opportunistic pirate who was quartermaster under the notorious Captain Flint.[1] Stevenson’s portrayal of Silver has greatly influenced the modern iconography of the pirate.[2]

Long John Silver has a parrot, named Captain Flint in honor—or mockery—of his former captain,[3] who generally perches on Silver’s shoulder, and is known to chatter pirate or seafaring phrases like “Pieces of Eight”, and “Stand by to go about”. Silver uses the parrot as another means of gaining Jim’s trust, by telling the boy all manner of exciting stories about the parrot’s buccaneer history. “‘Now that bird’, Silver would say, ‘is, maybe, two hundred years old, Hawkins—they lives forever mostly, and if anybody’s seen more wickedness it must be the devil himself. She’s sailed with England—the great pirate Cap’n England. She’s been at Madagascar, and at Malabar, and Surinam, and Providence, and Portobello… She was at the boarding of the Viceroy of the Indies out of Goa, she was, and to look at her you would think she was a baby’.”[4]