Military records generally used civil or Church of Ireland parishes, in this case the reference is to the civil parish of St. Thomas in Dublin’s north city. The attestation record provides a few additional details not included in the transcript :
The residence for Parick at the time he signed up is noted as ‘28 Upper Dominick Street’ which is off Dorset street/Bolton Street in the city.
In addition to his place of birth parish of St. Thomas in or near the town of Dublin in the county of Dublin, also confirms that Patrick lived in the same parish at the time he signed up, and also during the the previous 12 months.
The record also states his religious denomination as Catholic, but unfortunately does not include any notes regarding next of kin.
The office were Patrick's details were approved was the Curragh (Co. Kildare) like where he was initially based, and the only additional details of his time in the Army is that he was present in '98 and '99.
Upper Dominick Street is actually split between St. Mary’s and St. Michan’s civil parishes, but is close to St. Thomas. The Church of Ireland church for the parish of St. Thomas was on Marlborough Street just up from the the Catholic Pro-Cathedral.
The Catholic parishes covering the area around Upper Dominick Street would have been either St. Mary’s (Pro-Cathedral Marborough Street), St. Michan or possibly St. Agatha (North William Street), the same Catholic parishes would apply to St. Thomas. Given the timeframe a search for a civil birth might be the best option, if Patrick’s civil marriage record was available it should provide his father’s name and occupation to assist confirming or ruling out any matches. The registration district for the parts of the city around Upper Dominick Street and the civil parish of St. Thomas would be Dublin North.