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).
Your post had only sketchy info. The lack of info will likely result in high expense and low conclusiveness.
Lets work from what you know. If she emigrated in 1868, this strongly suggests that her son Patrick and daughters were born in Ireland. Assuming Patrick was born Co. Leitrim, I found only 6 B rcds within a relevant range: 2-1833, 1834, 1842, 1845, and 1846. None were in Carrigallen parish. The earlier rcds in this set are the more likely. It would be a long shot, at most 10% probability, that one of these rcds would reveal the correct parent's names.
If you know the daughter's names and approximate DOB, this could be circumstantial in helping to identify the parish for Patrick's B rcd, and perhaps the father's forename.
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.