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 Edward Mulligan B 1834 +/- 1 yr somewhere in Ireland with the father's forename unknown, I found 4 rcds: 1833 Meath, 1834 Wicklow, 1835 Cavan, and 1835 Longford. Father's forenames were William, Edward, Michael, and Michael respectively.
For Edward Bannon B 1812 ditto, I found 1 rcd in 1811 Co. Westmeath. Father's forename was also Edward.
Some of the above may be for your ancestors. You will need an exact DOB, parent(s) name(s), and/or placename to pick which, if any, is/are the right rcds.
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.