I'm posting an update to this question.
With some research, it appears that John and Hannah Bush are not Hannah's parents. The primary proof of this is ''New England Marriages Prior to 1700'' provides clues that Hannah is currently attached to the correct parents.
"Bush, Randall/Renold/Randolph & 1/wf Elizabeth ___?; b 1632 [sic?]; Cambridge<br/>
Bush, Randoll & 2/wf Susanna Lovett/Lovill?/Lowell, m/2 Thomas Hall b 1686; m cont 2 Sep 1678; Cambridge"