I usually 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 and almost non-existant before 1800. Many B/M/D events either weren't recorded, or the records haven't survived.
The last name and forenames suggest they were protestant. These records, especially of this vintage, are the least likely to have survived.
I found nothing conclusive while searching for their B records in the IFHF.
However members of the family seem exceptionally long lived. Two, at least, lived to late in the 19th century. Mandatory civil registration of B/M/Ds began in 1864. Civil records are kept by the General Register Office. Contact this office at WWW.GROIRELAND.IE. It might have some of the death certificates. Death certs often contain the parents' names, DOB, DOM, spouse, etc.