There are just 2 Parker names in the online telephone directory for Kerry.www.irishgenealogy.ie
If you can find a marriage record for Catherine you would have her father's name.
In 1853 Kerry underwent the Griffith Valuation and at that
time there was just 1 Parker of 17 to be in the Union of Kilarney. www.askaboutireland.ie
Thomas Parker, Parish of Kilbonane townland of Lahard,Poor Law Union of Kilarney (any town/village,parish,townland within 12 mi. of named town)
tenant of Ellen Dumas
15a Eugene O'Neil, house and land
15b William O'Neil, house and land
15c Thomas Parker, house and land
These three share 43 acres 3 roods and 11 perches
There are quite a few Parkers in the commercial directories but only these in 1846:
Benjamin Parker, shoemaker, Green Lane, Dingle
Lieut. Thomas Parker, Green Lane, Dingle
What were her children's names, the 2nd. male would likely be named after her father. www.censusfinder.com
Various information resourceswww.genealogy.nationalarchives.ie
Has 12 Parker's listed for the Tithe Applotments 1824-1839.