The cited source on the page for Jeremiah, claiming his parents as Thomas and Elizabeth (that is: Genealogy of Some of the Vail Family Descended from Jeremiah Vail at Salem, Mass, 1639, by Henry H Vail. Published in New York : DeVinne Press, 1902. 371p. Pg. 29. ) doesn't actually claim those parents. It says instead:
"JEREMIAH VAIL, b. probably in the west of England about 1618;"