According to the marriage record, William's parents were Thomas and Elizabeth Abby.
His name is spelled Abbey in the second marriage record, to Sarah Eaton. Also, the gravestone says Abbey. However, in other records, it is Abby.
In the 1865 MA State Census, the family is in Attleboro, MA: William Abby 40 Eng Operative, Ann 40 Eng wife, Elizabeth A. Abby 11 Eng, Edwin A. Farrar 19 RI, Charles E. Farrar 8 MA, Hurburt I. Farrar 6 MA. Elizabeth is William's daughter by a prior marriage, as the three boys are Ann's by her prior marriage.