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).
For Mary Tierney B 1845 +/- 2 yrs, Pallasgrean-Templebredon RC parish, with a father forenamed Pat(rick), I found 1 rcd in 1845.
For Mic(hael) B 1841 ditto, I found 1 rcd in 1841.
For Dan(iel) B 1835 ditto, I found 1 rcd in 1835.
For Marg(aret) B 1831 ditto, I found 1 rcd in 1832.
For Pat(rick) M 1829 +/- 2 yrs, Co. Limerick, I found 1 rcd in 1827, Pallasgrean-Templebredon RC parish.
For Mary Mulcahy M ditto, ditto.
All of the above are highly likely to be your ancestors. Views of the detailed rcds should confirm they are all related. Views of Patrick and Mary's M rcds SHOULD show their parents' names as well as other info useful in finding earlier generations.
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.