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 John Oates B 1823 +/- 5 yrs, Co. Roscommon, with a father named Francis, I found 1 record. A view of the detailed record (at a small fee), MAY show the mother's name and other information.
For Hannah/Anna Feeney B 1825 +/- 5 yrs, Co. Roscommon, with a father forenamed John, I found 0 records.