Statutory recording of births did not begin in Ireland till 1864. RC marriages were also registered from 1864 onwards and other denominations from April 1845 onwards. Prior to those dates you are dependant on parish records, where they survive. (Many parishes early records have been lost for various reasons).
You don’t say which denomination the family were. You need to try and establish that because the records are kept in different places. Some baptism and church marriage records are on-line but many are not. RC records are kept in the National Library, Dublin, though some are on-line on the following two sites. Church of Ireland records are in the RCB Library, Dublin.
Try this site for on-line records (but remember they don’t have all records):
You say the family were from Co. Cork but say a birth was registered in Castle Connell. As far as I can tell Castle Connell is in Limerick, not Cork. It’s in the parish of Stradbally. How certain are you of your information?