I searched the Irish Family History online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties. Records before 1863 are increasingly sparse: almost non-existant before 1800.
For the surname "Smith" B 1800 +/- 10 yrs, Co. Monaghan, I found 6 records. Three were forenamed John. None were forenamed Edward, David, Robert or Alice. One of the 3 MAY be for your ancestor, but it will be difficult to validate, especially because the other forenames didn't have any hits. Views of the detailed records often show the parents' names. Note - fees may apply.