Hello Kate,
There are three items you require when searching for Irish ancestors.
1. Denomination.
2. Parish
3. Townland.
You'll have to rely on Church records as civil registration in Ireland was not introduced until 1864. Non Catholic marriages were registered from 1845 onwards. When registration for BDMS started in 1864 some people didn't bother with the registration for a few years after that date. Keep your fingers crossed that church records still exist.
There are three listings for Parkinson on the 1852 extracts for Griffith's Valuations for Co. Kerry which are on the John Hayes failteromhat.com website.
John Parkinson, Castletown, Kilflyn. Kerry
Arthur Parkinson, Tullig, Kilfeighny. Kerry
Arthur Parkinson, Knocknakilly, Kilfeighny. Kerry
Christopher