I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
For Eleanor/Ellen Dolan with a father named Michael, I found 7 B rcds as possibles: 1860, 1864, 1865, 2-1872, 1877, and 1878. A view of the detailed B rcd should show whether the mother's name was Bridget. The B rcd should show the mother's maiden name. With these, you can look for matching M rcds for Eleanor's parents.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.