late 13c., "a parrot," from Old French papegai (12c.), from Spanish papagayo, from Arabic babagha', Persian babgha "parrot," possibly formed in an African or other non-Indo-European language and imitative of its cry. Ending probably assimilated in Western European languages to "jay" words (Old French jai, etc.).
Used of people in a complimentary sense (in allusion to beauty and rarity) from early 14c.; meaning "vain, talkative person" is first recorded 1520s. Obsolete figurative sense of "a target to shoot at" is explained by Cotgrave's 2nd sense definition: "also a woodden parrot (set up on the top of a steeple, high tree, or pole) whereat there is, in many parts of France, a generall shooting once euerie yeare; and an exemption, for all that yeare, from La Taille, obtained by him that strikes downe" all or part of the bird.