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 John Mulcahy B 1868 +/- 1 yr in Co. Cork with a father forenamed Michael, I found 0 rcds. Note 2 may apply.
For Michael Mulcahy B 1825 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 20 rcds - too many to list here.
For Michael Mulcahy M 1863 +/- 4 yrs in Co. Cork, I found 9 rcds. Note 2 may apply here as well.
Suggest you wait until the remainder of Cork rcds have been added to the database B4 searching further.
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.