I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
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 James Flavin B 1807 (exact), somewhere in Ireland, with the father's forename unknown, I found 1 rcds: 1807 Cork. Father's respective forename was Pat(rick).
For Cat(herine) Mahoney B 1820 (exact), somewhere in Ireland, with the father's forename unknown, I found 7 rcds: 6-Cork, and Limerick. Father's forenames were John, James, Pat(rick), Terence (Limerick), William, and Denis.
Since you have exact DOB, you should be able to pick which, if any, rcds are for your ancestors.
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.