"a small parrot," 1620s, from Spanish perquito; earlier English form parroket (1580s) is from French paroquet, from Old French paroquet (14c.), which is said by etymologists of French to be from Italian parrocchetto, literally "little priest," from parroco "parish priest," from Church Latin parochus (see parish), or from parrucchetto, diminutive of parrucca "peruke, periwig," in reference to the head plumage.
The Spanish form, meanwhile, is sometimes said to be a diminutive of Perico, a familiar form of Pedro "Peter," and the Old French word is likewise perhaps from or influenced by a diminutive of Pierre "Peter." "The relationship between the Sp. and It. forms cannot be settled until the chronology is known; prob. the name has been modified by popular etymology in one or both" [OED].