I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Your parameters are vague, so will do the best I can.
For Denis Enright B 1846 +/- 7 yrs, somewhere in Ireland, with a father forenamed John, I found 3 records, all in Co. Limerick.
For Mary Enright B ditto, I found 14 records, with 12 in Co. Limerick.
For John Enright B 1846 +/- 5 yrs, ditto, I found 20 records. However, 18 of the 20 were also in Limerick.
ASSUMING these B events were recorded and records survived, your highest probability is to look in Co. Limerick.
Suggest you try to get some more accurate DOBs for John, Mary and Denis. This would reduce the high number of false positives to a more manageable number. Then visit this database and do a more finely tuned search (note - nominal fees may apply. Try not to guess).
CAUTION - Irish records before 1863 are increasingly sparse. Many events weren't recorded, or the records haven't survived.