Welcome to Ireland Reaching Out and apologies for the delay in replying to your message.
As you can see from the link below, Kilaconenagh is a Civil Parish in Cork that includes Bere Island:
Records for the corresponding Roman Catholic Parish of Castletownbere begin in 1819. Dursey Island is in the Roman Catholic Parish of Castletownbere too, but in the Civil Parish of Kilnamanagh:
There are some places where you can search online, however these websites do not have complete collections and many do not cover the period after 1900:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
http://churchrecords.irishgenealogy.ie/churchrecords/ have begun a project to upload Irish church records to their site free of charge. However, so far only Counties Dublin, Kerry, Carlow and parts of Cork have been covered.
Other important sources include the Tithe Applotment Books and Griffith?s Valuation that you could check for Eugene, Timothy's father:
Tithe Applotment Books (1823-1837): http://titheapplotmentbooks.nationalarchives.ie/search/tab/index.jsp
Griffith?s Valuation (1847-1864): http://www.askaboutireland.ie/griffith-valuation/
Some other websites that may be helpful are:
The National Archives of Ireland: http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/ - you can search the 1901/1911 Census on the National Archives page too
Irish Times: http://www.irishtimes.com/ancestor/index.htm
Irish Genealogy Tool kit: http://www.irish-genealogy-toolkit.com/
You might also find these websites useful:
Remember to post any new information that you find here. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Best of luck with your search