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 parish, 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 Pat(rick) Daley B 1868 (exact) in Co. Limerick with a father forenamed Michael, I found 1 rcd, Cappamore parish. This MAY be for your ancestor. A view of the detailed rcd SHOULD show both parent names and other info.
For Mich(ael) Daley M 1864 +/- 4 yrs somewhere in Ireland, I found 35 rcds - too many to summarize here. However, 1 rcd was in Tipperary, Newport parish.
For Cath(erine) Shepard M 1864 somewhere in Ireland, I found 1 rcd in Co. Tipperary, Newport parish. These M rcds are LIKELY for your ancestors - no guarantees. A view of the detailed rcds will show whether the marriage was to each other. If so, the rcds SHOULD list their parents' names and other info. This will take you back at least one more generation.
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.