Statutory registration of births, deaths & RC marriages started in Ireland on 1.1.1864. James & Ann married before that and so, in those circumstances we generally rely on church records. However in this case Bodoney Lower RC parish doesn’t have any records earlier than 1865. So it’s unlikely that we will find any trace of not only their marriage but of either of their 2 childrens births in that parish.
You say that Patrick was born in 1864. That was the first year of statutory registration, so he ought to have a birth certificate even if there’s no baptism record. I searched for his birth but there's no trace of it. I’d conclude that either he was born before 1.1.1864 or the birth wasn’t registered.
I had a look at Griffiths Valuation for 1859 for the townland of Formil. I see just one SLAINE household (spelling). It was a farm headed by Peter Slaine. Plot 52a. He farmed in partnership with 2 others and between them they had 151 acres. He must be a contender for Ann’s father since it’s the only Slaine property in the townland. Also in the same townland was Michael Quin on plot 19, a 56 acre farm. If the Quinns also came from Formil, then again that might be his father.
Peter Slane died 13.7.1883 at Formil aged 70. His wife was still alive at that date. The informant was Michael Clarke, son in law, also of Formil. Here’s Michael in 1901:
I looked for Peter Slane’s wife’s death 1883-1901 but cannot see it in the statutory records.
Probate abstract from the PRONI wills site:
Clarke Michael of Formil county Tyrone farmer and ex-school teacher died 18 December 1922 Probate Londonderry 5 March to Margaret Clarke widow. Effects £183 15s. 3d.
The Slane family lived next door:
I looked to see if the families were farming in that townland in the tithes in 1830 but they are not listed:
Slane probate abstracts:
Administration of the Estate of James Slane late of Formil County Tyrone Farmer who died 27 May 1912 granted at Londonderry to Susan Slane the Widow
Slane Patrick of Formil county Tyrone farmer died 8 January 1928 Administration Londonderry 20 December to Catherine Slane spinster. Effects £42 10s.
Slane Margaret of Formil county Tyrone widow died 4 June 1939 Probate Londonderry 26 January to Patrick Slane farmer. Effects £211 3s. 5d.