Here we have the parents listed as Anthony Convers and Clemence Spady. The link in the About Me section to the 1998 paper, "The Road to Royalty is Broken", by Kurtz, is not an operable link, so I am unable to locate that work. However, the current info on the Converse Family site, in an article by Harold Whitmore, shows the father of Edward as Allen Convers, brother of Anthony, with mother's name unknown. (https://conversefamily.com/deacon-edward-convers.)
In a 1999 article, "English Origins and Ancestry of the Parker Brothers of Massachusetts, and Their Probable Aunt, Sarah Parker, wife of Edward Converse," [NEHGR 153;81-89] Douglas Richardson is in agreement with Allen as the father, and calls Edward a brother of Allen Jr.
Is there any subsequent information to validate the choice of Anthony as the father? I note that Erica Howton has a curator's note dated 2012 affirming Anthony as the probable father.