Since I posted my initial inquiry, I have additional information on Michael and Nora O'Connell Dynan. I still have not found information on where they originated from in Ireland or the names of their parents. A preliminary search done through the Clare Heritage Centre found no leads.
Michael was born about 1822, died 1870. Hanorah (Nora) was born 1827 (County Clare), died 1911. They came to the U.S about 1847, possibly from Dublin) and settled in Illinois where their surname became Dining. Except for Michael's brother Martin Dynan who went to Australia and brother Dennis who died in Dublin in a religious riot, Michael's siblings and spouses (a sister married to Coleman; a sister married to Kilfoil, a sister Sydney married to John Comer, a sister Catherine married to Thomas Carmody, and a brother William) emigrated to and settled in IL. Hanorah's siblings and spouses also emigrated to the U.S.-- her sister Margaret married to Dennis Mulqueen (?) and half sister Mary married to Patrick McNamarra (?). Any leads on some of these ancestors are welcomed.