The Mayflower probably carried as many as twenty or more average seamen when it sailed from Southampton in 1620, but history only records a couple of their names. The Mayflower in known to have carried four quartermasters, gunners, a cook, a carpenter and boatswain, all their names went unrecorded however there was a crew member from the Parker family.

http://members.aol.com/calebj/seamen.html

John or Thomas Parker

The early 20th century Mayflower scholar Charles Banks identifies this man as a crewmember. This is based on a 1750 Superior Court deposition of one John Phillips. He stated that when he was twelve his father told him ‘that Sagadahock had the name of Parkers Island given to it from one Thomas Parker who as said John Phillips was Informed was Mate of the first ship that came from England with the Plymouth People and was uncle to my father John Phillips and father to Thomas Parker who then lived on Parkers Island.’. Thomas Parker’s father was John Parker, and so according to Banks must have been a mate on the Mayflower.

Charles Edward Banks, English Ancestry & Homes of the Pilgrim Fathers 1929