The record of the marriage of Michael Dillon and Mary Murnane is on Roots Ireland (subscription site) The details are as follows.
Date. 21 June 1861.
Husbands name: Michael Dillon.
Wife's name: Mary Murnane.
Denomination: Roman Catholic.
Wittnesses: Michael Murnane and Anne Leary.
Unfortunately it does not give the first names of the fathers of the bride and groom.
Michael and Mary had 5 children.
Margaret approx 1862
Hanorah 1867. Address: Gragane.
John 1870. Gragane.
Michael 1872. Caherline.
Bridget 1875. Gragane.
Margaret married James Hickey in 1899. This is the record.
I did not find a record of marriages for Bridget, Michael or John.
Hanorah turns up in the census living with her sister Margaret in 1901.
And again in 1911.
Michael Dillon senior died in 1886.
His wife Mary Murnane died in 1897.
The wittness on the marriage of Michael and Mary, Michael Murnane was most likely the brother of Mary. I searched for a marriage for him and found this.
The address would appear to confirm this fact. And now you can see their father was called Michael Murnane also.
I haven't found a record of his marriage to find out the mother's name but I will have another look later.
I hope this is of help to you.