We have conflicting information for the parents. The about me sources several places that state John Conyers & Margaret Darcy. The one Woodman presented supports Christopher Conyers and Margaret Waddiley. According to the sources, John Conyer is Christopher's son, through his first marriage. So the question is if Christopher is Alice's father or grandfather. If the birth date of Alice is accurate, it would seem to make more sense if she was Christopher's child.
"If you go to this page:
http://wc.rootsweb.ancestry.com/cgi-bin/igm.cgi?op=GET&db=a......
it puts in an extra generation, a Christopher Conyers. And it does seem to be unusually well-sourced (including a heraldic visitation) although I have not checked the sources myself.
Sincerely,
Woodman Mark Lowes Dickinson, OBE"