If your Mary Elizabeth was born in the U.S., perhaps her parents married here. In any case, either the marriage record or the death record could give you Daniel's parents' names. I have Carrolls from Kildorrery, Cork.
My family came from Kildorrery, Cork. Denis Carroll married Margaret Kennedy in 1813. Their children that I know of included Peggy, Ellen, Denis, V. Ann (the V. probably for Veronica), Edmond, Mary, Catherine, and Anthony. Daniel was also a name connected with this family.
I was wondering if any of these names were in your family or where in Cork your family might have been from.