I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc for MOST Irish counties. Records before 1863 are increasingly sparse and almost non-existant before 1800. Protestant records are the least likely to have survived.
For Ed(ward) Brannagan B 1823 +/- 5 yrs, Co. Wicklow, with the parents' names unknown, I found 1 record: 1821. Assuming this MAY be for your ancestor, you can view the detailed record (usually for a small fee). This SHOULD show the parents' names as well as other info such as parish, etc.
In general for the surname Brannagan, Co. Wicklow, there are 53 B and 33 M records for ALL dates. Throughout ALL Ireland, there are 4108 B and 2100 M records, again for ALL dates.