Fermoy, Cork

H Eleanor,

This is the Catholic Parish books. You should find him in here.
Fermoy, Cloyne - Catholic Parish Registers at the NLI

Thanks very much.   I noticed only one illegitimate child being baptised in that period.   Are such children less likely to appear?

Also, is it possible to determine the civil parish name?

Hi Eleanor, 
St. Patrick's is the Fermoy Church......this link says it was completed in the 1820's, so this must be where the Baptisms took place.
Fundraiser by the Restoration Fund committee : St. Patrick's Church, Fermoy, Restoration (gofundme.com)
This is a more modern one.
Churches | Fermoy Parish Website

Did you see the actual Baptism record?
Some times they give a townland name, if they did then google/search the townland name and Fermoy and it would show on a map.

You say...
I noticed only one illegitimate child being baptised in that period.
If the record states both parents, it means that the child was conceived before the marriage, within 9 months.
If parents were not married then it would only state the Mothers name and illegitimate where the fathers name would be. 
Margot