I searched the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D records for MOST Irish counties.
For Th(omas) Halfpenny B 1801 +/- 5 yrs, somewhere in Ireland, with his parents' names unknown, I found 11 records. Of the 11 one is in Limerick, the other in Tipperary. The others are elsewhere.
Note 1. at this early date, few B/M/D events, especially for the poor, were rcd'd, and many which were haven't survived. Note 2. Portions of the surviving B/M/D records for Tipperary are contained in a church controlled database. Viewing these records requires a substantial fee. Note 3. The chance that either of the records I found are for YOUR ancestor is very small, perhaps 5%.
Suggest you visit the database and search for Thomas and Mary's other offspring. There is a better chance that one or more of these records may have survived.