I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 20 M church/civil/census B/M/D records for MOST Irish counties, including Armagh. Records before 1863 are increasingly sparse - almost non-existant by 1800. Protestant records are the LEAST likely to have survived.
For Cha(rles) Carberry B 1866 (exact), Co. Armagh, with the parents' names unknown, I found 2 records. One, the other, or neither may be for your ancestor. A view of the detailed record(s) (at a small fee) SHOULD show the parents' names and sometimes other details.
Will leave it to you to search the the other offspring.