I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
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).
Note 4 - Microfilms of parish church rcds are kept by the National Library of Ireland (NLI).
For Jeremiah O'Shea B ????, somewhere in Ireland (not just Cork) with a father forenamed Cornelius, I found 9 records: 1809 Cork, 1820 Cork, 1821 Cork, 1840 Cork, 1851 Cork, 1851 Cork, 1858 Cork, 1869 Cork, 1877 Cork, and 1888 Cork. Note - Since only abt 1/2 of Corks records have been transcribed, there may be an equal number which aren't available yet.
The forenames Jeremiah and Cornelius are widely used in Munster Province (Cork, Kerry, Limerick, Tipperary, etc. Of the rougly 14,000 O'Shea B rcds in Ireland, nearly 4,000 are in Co. Cork.
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.