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).
For Mich(ael) Fogarty jr B 1853 +/- 1 yr somewhere in Ireland with a father forenamed Pat(rick), I found 2 rcds: 1853 Tipperary (Youghalarra parish) and 1854 Kilkenny (Johnstown parish).
For Julia Fogarty B 1858 ditto, I found 0 rcds.
For James Fogarty B 1862 ditto, I found 2 rcds in 1864 Limerick, one civil, one church, likely duplicates of the same event.
For Mary Fogarty B 1868 +/- 2 yrs ditto, I found 4 rcds: 1866 Tipperary, 1866 Limerick, 1870 Kilkenny, & 1870 Dublin.
For Pat(rick) Fogarty Sr M 1851 +/- 1 yr somewhere in Ireland, I found 1 rcd in 1853 Limerick.
Because I can't find a consistent parish and DOB for the offspring, my results are inconclusive. It is doubtful if these are for your ancestors. Suggest you wait until the database is more complete. This may take some time.
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.