You should get DOB/DOM for Matthew/Julia from the censuses and/or naturalization papers. The DOBs for the children would also have helped.
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 Mat(hew), La(urence), and Paul Tierney with a father forenamed Mat(hew), I found 0 rcds, or inconsistent rcds. The children were likely B in America.
For Mat(hew) Tierney M 1880-1906, somewhere in Ireland, I found 3 rcds: 1881 Tyrone, 1899 Cavan, and 1901 Tipperary. I suspect Mathew and Julia married in America.
For Mat(hew) Tierney B 1880 +/- 4 yrs, somewhere in Ireland, with the father's forename unknown, I found 4 rcds: 2-1877 Armagh, 1878 Tyrone, 2-1879 Antrim, and 1880 Limerick.
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.