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 county, 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 Peter Gavin B 1859 +/- 4 yrs in Co. Galway with a father forenamed Cornelius, I found 0 rcds. 5 of 31 RC parishes in Galway (east) were NOT keeping B rcds in 1859.
For Marg(aret) Whelan B 1860 ditto with a father forenamed Pat(rick), I found 1 rcd in 1861, Athenry RC parish. A view of the detailed B rcd may show that the mother's forename was Bri(dget). If so, it MAY be for your ancestor - Bri(dget) was a very common forename.
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.