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).
Note 4 - Microfilms of parish church rcds are kept by the National Library of Ireland (NLI).
For James Flavin B 1807 (exact), somewhere in Ireland, with the father's forename unknown, I found 1 rcd, Co. Cork, Castlelyons RC parish. Father's forename was Patrick.
For Cat(herine) Mahoney B 1820 ditto, I found 7 rcds: 6-Cork, and 1-Limerick. Parishes were Castletownroche, Aghada, Cloyne, 2-Doneraile, Macroom, and Galbally & Aherlow RC parishes. Father's respective forenames were James, Edmond, William, John, Patrick, Denis, and Terence.
You will need exact DOBs, parent(s) name(s), and/or placename(s) to pick which, if any, are for your ancestors.
For immigration rcds, google "castlegarden" and/or NARA.
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.