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 An(ne) Fahy B 1876 (exact), Co. Galway east, with a father forenamed Pat(rick), I found 1 civil B rcd in Gort Civil Parish. A view of the detailed rcd should confirm the mother's forename/surname. This will help validate the rcds for the siblings.
For Sara(h) Fahy B ???? ditto, I found 1 rcd in 1871, Gort CP.
For Cath(erine)(Kate) Fahy B ???? ditto, I found 4 rcds: 1868, 1872, and 2-1873.
For Mary/Maria Hansberry M 1867 +/- 4 yrs, Galway east, I found 4 rcds: 2-1865, 1866, and 1870.
For Pat(rick) Fahy M ditto, I found 7 rcds: 1864, 1865, 3-1866, and 2-1868. The 1866 rcds are the only ones in common. However, a screen at the parish level of detail revealed 0 matches. They may have married outside of Galway east, or at a date outside of my search range. Will let you chase these rabbits.
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.