I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse and almost non-existant before 1800.
For Thomas Conroy B 1858 +/- 1 yr, somewhere in Ireland, with a father forenamed Edward, and a mother named SARA Deegan, I found 1 record: 1857, Co. Laois. This matches your criteria exactly. It is highly likely to be for your ancestor.
For Ed(ward) Conroy and SARAH Deegan M 1852 +/- 5 yrs, somewhere in Ireland, with the parents' names unknown, I found 1 record: 1846, also in Laois. A view (usually for a small fee) of the detailed record MAY reveal their parents' names as well as other details.