Hi, to answer your question.I only have found Elizabeth Cator's parents names but not sure for the fact. I am still doing some more researching on that.
I am a direct descendant of John Bickford, b. 1607, ENG married to Elizabeth Cator, b. @ 1620 ENG.
Elizabeth Cator's parents are Richard Cator, b. @ 1590 ENG married to Mary Ricord (married 16 Apr 1672 Portsmouth, NH) That is the only info I have of them. They had two children:Elizabeth, b. @ 1620 ENG & Richard Cator, b. @ 1622 ENGI got the info from www.ancestry.com
I will email you directly for more info on the family tree if you are interested.