There are several dates of birth floating around for Richard Parke, none of which have a proof. There are some christenings in various places in England, but no proof that it is the right person. In a Parke family history there is a statement that says he could be the son of Thomas Parke of the Isle of Ely. The one thing that makes me give this more thought is that Richard named his first son "Thomas."
Thanks for your help!
Susie