Further information, found in Frank Sylvester Parks, "Genealogy of the Parke Families of Connecticut", 1906, page 38, William was married three times, to Hannah Frink, Hannah Plimpton, and Mary ___.
William's parents should be Thomas Parke-8 (merged with Parke-368) and Dorothy Thompson-563 (merged with Thompson-7648). And there's another profile for him, Parke-174, that shows incorrectly that he died in 1654 (as an infant).