I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse and al most non-existant prior to 1800. Many early B/M/D events weren't recorded, or the records didn't survive.
For Michael Moran B 1813 +/- 2 yrs, Co. Kildare, with the parents' names unknown, I found 2 records: both in 1815. One, if either, MAY be for this ancestor. Will leave it to you to view the detailed records (usually at a small fee) to pursue this. The view SHOULD show the parents' names.
These clues MAY be useful. Good luck and merry Xmas.