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).
All the children appear to have been B somewhere other than Cooley parish, Co. Louth. Will let you find them.
For Mic(hael) Deighan M 1857 (exact), Cooley RC parish, Co. Louth, I found 1 rcd.
For Mary O'Neill M ditto, ditto. This is a match. The rcds are HIGHLY likely to be for your ancestors. A view of the detailed rcd should confirm this and reveal their parents' names. This info is useful in identifying and verifying earlier B/M rcds.
For Mic(hael) Deighan B 1832 +/- 5 yrs, somewhere in Ireland, with the father's forename unknown, I found 14 rcds: 1828 Wicklow, 1829 Armagh, 1830 Galway, 1831 Wicklow, 1832 Tipperary, 1833 Kilkenny, 1833 Westmeath, 1834 Kilkenny, 1835 Kildare, 1835 Limerick, 2-1835 Cork, 1836 Kilkenny, and 1837 Kilkenny.
For Mary O'Neill B 1836 ditto, I found 656 rcds, with 28 in Co. Louth, and 9 in Cooley RC parish. For Cooley, the father's forenames were John, Mic(hael), 4-Pat, Philip, Henry, and Owen.
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.