I don't disagree that Anne was married to Patrick; it just might be that she was the second wife, so that Edward's mother might be somebody else. My grandmother was also married at 15, but given the ages, Ann would have had to have been anywhere from 8 to 13 if she gave birth to Edward.
Note in the 1850 Census, Anne and Patrick have two children living at home -- one aged 18 and one aged 5 -- with none in between; this long time between the two children suggests a second marriage.
These same family trees also have Anne living in New York in 1874, the widow of Patrick; when his tombstone says he died in Iowa in 1879.Just saying that these online trees can be a mess, and shouldn't be used as any kind of conclusive evidence.
Another online tree has Edward and Patrick as the sons of Patrick, with an unknown mother; and showing Patrick's second marriage to Ann Bingham with children Ellen & Isabella. This is much more likely to be the case.