I'm alsoa desc. of John Bickford + Eliz. Cator by son Thomas. If the date you have for the marriage of Mary Picord is correct, I doubt that it could have been to Richard Cator as I believe he died prior to 1664. Also Eliz. was b. Abt 1620. The 1672 date sounds very much like the death date of Eliz. There also seems to be 2 more siblings for Eliz. Edward b. Abt1624 and Mary b. Abt 1626.
The only other info. re parents of Eliz. et al that I've found only says that they are the ch. of Richard's first sp. (?) Anyone with conflicting or additional info please feel free to e-mail me directly.