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 Michael Sheehan B 1839 +/- 2 yrs in Co. Cork with the father's forename unknown, I found 24 rcds.
For Julia McNamara B 1843 +/- 4 yrs ditto, I found 2 rcds: 1840 and 1847.
For Michael Sheehan M 1861 +/- 3 yr in Cork, I found 0 M rcds.
For Julia McNamara M ditto, I found 0 M rcds.
For Cornelius Sheehan B 1867 +/- 1 yr in Co. Cork with the father forenamed Michael, I found 1 B rcd in Mallow parish in 1867.
For Mary Sheehan B 1869 +/- 1 yr in Mallow parish, Co. Cork, I found 3 rcds: 1869, and 2-1870.
For Hannah Sheehan B 1871 ditto, I found 0 rcds.
For Edward Sheehan B 1865 ditto, I found 0 rcds.
These results are very poor. Your info is really poor, or these events occurred in the part of Cork which hasn't been digitized yet. I think the latter is the case.
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.