Do you have any dates for your ancestors?
There are Roman Catholic church records available for the Cory City parish/diocese of St. Finbarr's. These date from 1756 for baptismal records and 1753 for marriage records. You can find these at the National Library of Ireland, B.1776-1856, Pos. 4778; B. 1857-1880, Pos. 4779; M.1775-1810, Pos. 4779; M.1810-1880, Pos. 47801. If you have any difficulty, you could try writing to the parish priest for more assistance.
Civil registration records are available from the General Register Office (GRO). These start from 1864 however. You can access the website here: http://www.groireland.ie/research.htm
Have you tried checking the Ellis Island records (www.ellisisland.org) or the Castle Garden records (www.castlegarden.org)? Generally, more information was given at the port of arrival rather than the port of departure. If you knew which city they arrived at, this could be a good place to find more information, and perhaps even find out an exact place of origin. How do you know that St. Finbarr's is the correct parish? For your information, there are many parishes within Cork city and these will have different records.
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64). Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at.
Please make sure you link anyone else in your family who is interested in their Irish heritage to our site - and indeed anyone else you know of Irish heritage.
Genealogist (Ireland XO)