I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
For Phillip Fines M 1867 (exact) in all Ireland, I found 2 M rcds, both in Armagh, one civil, one church.
For Ann McGaul M 1867 (exact) in all Ireland, I found 2 M rcds in Co. Armagh: one civil, the other church, likely duplicates.
From the M rcd, you should find their parents' names. This MAY let you push your geneology back at least 1 more generation.
For Mary Ellen Fines B 1880 (exact), ditto, I found 1 B rcd.
For John Fines B 1872 (exact) ditto, I found 0 B rcds.
B/M/D registration was civil after 1863. Civil rcds are kept by the General Register Office for the Republic of Ireland and by the Public Record Office for Northern Ireland. Google "PRONI" for the latter.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.