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 Mich(ael) Malloy B 1874 +/- 2 yrs, Ardara CR district, Co. Donegal, I found 1 rcd in 1874. The father's forename was Patrick.
For Pat(rick) Malloy B 1872 +/- 4 yrs, ditto, I found 5 rcds: 3-1870, 1873, and 1875. Father's forenames were Edward, Lanty, James, James, and William respectively.
For Patrick John Malloy B ????, Ardara CR district, Co. Donegal, I found 2 rcds: 1897, and 1916. The 1916 date is outside your range. The 1897 date appears to be correct. Father's forename for this rcd was Bernard.
For Bernard Malloy B ????, ditto, I found 2 rcds: 1865, and 1905. Father's respective forenames were Pat(rick) and James.
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.