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 Mary Doolan B 1861 +/- 4 yrs, somewhere in Ireland, with a father forenamed Mic(hael), I found 2 rcds: 1861 Roscommon, and 1864 Tipperary.
For Mic(hael) Doolan M 1857 +/- 4 yrs, somewhere in Ireland, I found 3 rcds: 1856 Tipperary, 1860 Cork, and 1861 Tipperary.
For Br(idget) Cassidy M ditto, Tipperary and Cork, I found 0 rcds. This infers the above results may all be false positives, or the rcds haven't been transcribed yet.
For Mic(hael) Doolan B 1836 +/- 4 yrs, somewhere in Ireland, with the father's forename unknown, I found 10 rcds: 1832 Dublin, 1835 Dublin, 1835 Westmeath, 1836 Limerick, 1836 Kildare, 1837 Roscommon, 1838 Limerick, 1838 Wicklow, 1838 Longford, and 1840 Cork. Father's respective forenames were John, Michael, Simon, Michael, John, Michael, Patrick, John, William, and William.
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.