I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse. By 1800, they are almost non-existant. Many B/M/D events either weren't recorded, or haven't survived.
For James McKernan B 1840 (exact), Co. Cavan, with the parents' names unknown, I found 1 record. However, the father's name either wasn't recorded, or at least wasn't F(rancis). A view (usually at a small fee) of the detailed record MAY show the parents' names and other lesser details.