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 Martin Moran M 1876 +/- 1 yr, somewhere in Ireland, I found 10 rcds: 2-1876 Leitrim, 3-1876 Mayo*, 2-1877 Galway*, 1877 Westmeath, and 2-1877 Mayo.
For Mary Farrell M ditto, I found 41 rcds - too many to easily summarize here. However, POTENTIAL matching rcds are indicated in the yr/cnty combinations by asterisks above.
For Martin Moran B 1843 +/- 4 yrs, somewhere in Ireland, with a father forenamed John, I found 1 rcd, Elphin RC parish, Co. Roscommon.
I also found all of the other forenamed children in a short span of years around 1843 with a father named James in Elphin parish.
For John Moran M 1843 +/- 5 yrs, somewhere in Ireland, I found 87 rcds - waaaay to many to summarize. However, 14 are in either Leitrim, or Roscommon. None are in Elphin RC parish.
For Do(rah) Flanagan M ditto, I found 2 rcds, 1842, for Dorothy in Co. Roscommon, both in Knockcroghery RC parish. Neither matches any of the rcds for John. It may be that this rcd didn't survive, or hasn't been transcribed.
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.