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) Fallon B 1820 +/- 1 yrs, Co. Roscommon, with the parents' names unknown, I found 2 records.
For Mary Fallon B 1823 (exact), ditto, I found 9 records.
For Ed(ward) Fallon B 1829 +/- 1 yr, ditto, I found 1 record.
For Mi(chael) Fallon B 1833 (exact), ditto, I found 6 records.
For Hon(or) Fallon B 1843 (exact), ditto, I found 2 records.
For John Fallon B 1831, ditto, I found 4 records.
There are obviously several Fallon families in the above. Will leave it to you to view the detailed records (at small fees) to find the respective parents.