I research the Irish Family History Foundation's database. It contains 12,700,000 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 30%.
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 George Mulloy/Molloy B 1815 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 1 rcd in 1814, Co. Dublin. However, the father's forename isn't John. The only rcds for this name and the father's name of John in Co. Mayo are in 1872. This is WAAAY too far from your timeline. It is unlikely if additional early rcds will be added to the Mayo database.
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.