I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties.
For Dan(iel) Lee B 1840 +/- 2 yrs, somewhere in Ireland, with parents named John and mother HONORA Hickey, I found 1 record, Co. Tipperary. This matches your criteria well. It is likely for your ancestors. A view (at a small fee) of the detailed record should reveal additional info.
You can search this database further for B records of the siblings and marriage record of the parents. The marriage record, if it exists, MAY list the names of their parents.