I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties, including Sligo. Records before 1863 are increasingly sparse: almost non-existant before 1800. Protestant records are scarce for all dates.
For a Mi(...) Armstrong B 1826 +/- 2 yrs, somewhere in Ireland, with the parents' names unknown, I found 2 records. Neither was in Co. Sligo.
For Eliza Ewing B 1831 (exact), ditto, I found 1 record in Co. Sligo. Parent's forenames appear to be John and Mary. The exact DOB should allow you to validate the record. If so, a view of the detailed record (fees may apply) MAY show the surnames for the parents as well as lesser details.