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 Michael Darmody B 1859 +/- 4 yrs, somewhere in Ireland, with the father's forename unknown, I found 11 rcds, but none had a father forenamed Thomas.
For Thomas Darmody M 1854 +/- 4 yrs, somewhere in Ireland, I found 4 rcds: 1850 Tipperary, 1853 Kilkenny, 1856 Westmeath, 1856 Longford, and 1858 Tipperary.
For Jane Quigley M ditto, I found 2 rcds: 1855 and 1858, both in Co. Derry. This is a NO match.
Since your post states Dublin as a potential origin, this causes a problem. Co. Dublin rcds are in the database; Dublin City rcds aren't. You should contact the appropriate denomination's diocese for these records.
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.