mid-13c., "enclosed preserve for beasts of the chase," from Old French parc "enclosed wood or heath land used as a game preserve" (12c.), probably ultimately from West Germanic *parruk "enclosed tract of land" (source also of Old English pearruc, root of paddock (n.2), Old High German pfarrih "fencing about, enclosure," German pferch "fold for sheep," Dutch park).
Internal evidence suggests the West Germanic word is pre-4c. and originally meant the fencing, not the place enclosed. Found also in Medieval Latin as parricus "enclosure, park" (8c.), which likely is the direct source of the Old French word, as well as Italian parco, Spanish parque, etc. Some claim the Medieval Latin word as the source of the West Germanic, but the reverse seems more likely. Some later senses in English represent later borrowings from French. OED discounts notion of a Celtic origin. Welsh parc, Gaelic pairc are from English.
Meaning "enclosed lot in or near a town, for public recreation" is first attested 1660s, originally in reference to London; the sense evolution is via royal parks in the original, hunting sense being overrun by the growth of London and being opened to the public. Applied to sporting fields in American English from 1867.
New York's Park Avenue as an adjective meaning "luxurious and fashionable" (1956) was preceded in the same sense by London's Park Lane (1880). As a surname, Parker "keeper of a park" is attested in English from mid-12c. As a vehicle transmission gear, park (n.) is attested from 1949.