I research 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.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For James Mullarkey B 1885 +/- 4 yrs, Co. Mayo, with the father's forename unknown, I found 7 rcds: 2-1883, 2-1884, 2-1887, and 1889. Father's respective forenames were John, John, James, James, David, David, and James. Note - the pattern indicates half are civil duplicates of church rcds. The civil districts were Ballyhaunis (83, 84, and 87) and Knocknalower (89).
For Nora/Honor/Honora McMenamin B 1887 ditto, I found 4 rcds: 1883, 2-1887, and 1888. Father's forenames were Michael, Hugh, Hugh, and Anthony respectively. The 2 Hughs are likely church/civil duplicates.
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.