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 - The National Library of Ireland (NLI) has microfilms of parish church rcds.
For John Fitzpatrick B 1812 (exact), somewhere in Ireland, with the father's forename unknown, I found 8 rcds: 2-Dublin, Wicklow, Down, Tipperary, 2-Kilkenny, and Cork. You have an exact DOB. This will help you pick which, if any, rcd is the right one.
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.