I would normally expect that John's father's name to be Peter if that is what he names his eldest son (although it's not set in stone - that was part of the traditional Irish naming pattern).
Have you found his marriage to Ann? Was that in Ireland or England? If it was in England you could order the marriage cert and obtain his father's name.
If in Ireland - before 1864 you will need the parish record, which often gives the fathers names.
In addition -- if they had children in England - you can obtain Ann's maiden name from the Index at the General Register Office from the child's birth entry.
That would help in finding the correct marriage.
