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 Felix TierneyB 1795 +/- 4 yrs somewhere in Ireland with the father's forename unknown, the earliest B rcd I found was for 1827 in Cavan.
For Mary Malley B ditto, I found 9 rcds: 1792 Galway, 3-1793 Galway, 1794 Roscommon, 1796 Galway, 1796 Antrim, 1794 Louth, and 1799 Antrim.
For Felix Tierney M 1820 +/- 4 yrs somewhere in Ireland, I found 0 rcds. Without Felix's rcds there is no sense looking for those for Mary.
You MIGHT be able to find the children's B rcds if you have their forenames and DOB.
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.