I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church/civil/census/etc B/M/D records for MOST Irish counties. Records before 1863 become sparse - almost non-existant before 1800. Protestant records were the least likely to survive.
For Pat(rick) Mulry B 1803 +/- 5 yrs, Co. Roscommon, with the parents' names unknown, I found ZERO records.
For the surname MULRY, Co. Roscommon, in the range 1820 +/- 20 yrs, I found 76 B and 38 M records.