It looks like your George married Catherine at St Pancras Church of England (Anglican) in London on 1st August 1870 (London Metropolitan Archives/Ancestry). At the time of marriage George is recorded as a Bricklayer, and his father Barnaby as a foreman, the bride's father Cornelius is shown as a labourer, both the bride & groom are sown as 'full age' so 21 or over. Historic civil marriage records for England/Wales (same as Ireland at that time) do not include mother's names.
It's worth double checking the full details of records you find on Ancestry - not all the records they show originate from primary sources, e.g. a number of the results I see relating to George & Catherine are user submitted trees, which are not always accurate.
To search for vital records, e.g. baptisms & marriages, in Ireland the first step would be to establish the relevant Church of Ireland (also Anglican and equivalent to the Church of England) parish or parishes, and then check to see what records are available. Relatively few Church of Ireland records are currently available online for Co. Cork at the moment.
The town of Dunmanway is in Fanlobbus civil parish, and according to the latest (Sept. 2019) parish list from the RCB LIbrary, the early records for this parish have not survived. The list also notes that some original records may still be held by the parish, but dont give any details. Some later records for the parish are available in the National Archives (Dublin), e.g. baptisms 1855-1871, burials 1855-1871 and list of Parishioners dated c1848.
Dromdrasdil as mentioned in the original post is a townland within Fanlobbus civil parish.
A search of Fanlobbus parish on Griffith's Primary Valuation (dated 1852 for this area) shows a number of Chambers properties (just under 30), but no sign of a Barnaby or Barnabus, so possibly was deceased, lived in some shared or other ineligible accommodation or a building provided by an employer or lived somewhere else at the time. THe earlier Tithe Applotment records dated c1826 also show a number of Chamber properties eligible for these payments, but no sign of a Barnabus with property liable for this form of tax.