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 parish, 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).
Co. Waterford rcds aren't in this database yet. Suggest you wait until they are. This may take some time; be patient.
For Cath(erine) Starrs B 1889 in Omagh RD, Co. Tyrone with a father forenamed Fr(ancis), I found 1 rcd.
For Fr(ancis) Starrs M 1888 +/- 3 yrs somewhere in Ireland, I found 2 rcds in 1886, Co. Armagh. 1 of these MAY be for your ancestor - no guarantee.
For Fr(ancis) Starrs B 1864 +/- 4 yrs somewhere in Ireland with a father's forename unknown, I found 4 rcds: 2-1862 and 2-1867, all in Co. Tyrone. However, none of them had a father forenamed Peter.
For Peter Starrs B ????, I found 0 rcds within a relevant range of yrs.
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.