I research the Irish Family History Foundation's database. It contains 12,700,000 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 30%.
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 Pat Mulrooney B 1867 +/- 4 yrs, Co. Mayo, with a father forenamed Charles, I found 1 rcd, 1866, Burriscarra RC parish. This is likely for your ancestor. A view of the detailed rcd SHOULD show the parent's names. This will help validate the parent's marriage rcds, and sibling's B rcds.
Potential siblings are Anne 1865, Bridget 1873, and James 1869.
For Charles Mulrooney M 1862 +/- 2 yrs, Co. Mayo, I found 1 rcd in 1860, Burriscarra RC parish. A view of the detailed rcd SHOULD show the bride's name and both parent's names as well as other info. This will help validate earlier B/M rcds.
For Charles Mulrooney B 1838 +/- 4 yrs, Co. Mayo, I found 0 rcds within a relevant range. Note - this date is early for Irish records, especially for the western counties. It is likely the rcd didn't survive. There are also NO rcds for this name at this early date in the rest of Ireland.
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.