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 Barbara Faherty B 1895 +/- 4 yrs, Co. Galway, with a father forenamed Thomas, I found 1 rcd in Kilcummin Oughtergard RC parish. This fits your criteria exactly. It is likely for your ancestor.
For Thomas Faherty M 1891 +/- 4 yrs, Co. Galway, I found 2 rcds: 1891, and 1894, Kilcummin Oughtergard parish.
For Mary Molloy M ditto, I found 1 rcd in 1891, ditto.
These are highly likely to be for your ancestors. A view of the 1891 rcd SHOULD show they married each other. If so, this validates the rcds. The view also SHOULD reveal their parents' names and other info.
By now you should have duplicated my search of this database. Will leave it to you to search for the B rcds of the other offspring.
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.