Susan:
Death certificates do not show parents names. Many times the record will show the relationship of the informant to the deceased. Usually, son or daughter.
John Mulligan and Margaret Dolan married January 21 1863 in Drumlumman RC parish in Co. Cavan. Granard in Longford and Drumlumman in Cavan border. Marriages usually were held in the bride's parish.
Civil registration of marriages started in 1864 and showed the names of the fathers of the bride and groom. So you are one year off from getting a civil record.
I searched on the subscription site Roots Ireland for John Mulligan's in Granard. There was one in 1832 but the next record was in 1845. So not possible to give you a possible lead.
Roger McDonnell