If you're looking for Roman Catholic records and don't know what parish to contact, the best place to start is usually the diocesan archives. In Connecticut, there are three Roman Catholic dioceses: Hartford, Norwich, and Bridgeport. Information for the diocesan archives in Hartford is available on their website. Norwich is a little harder to find: visit the diocesan website and click on the Diocesan Archivist information.
However, be aware... Norwich is a relatively new diocese. It's only existed since 1953. Before that, you need to go to Hartford any records. Happy research!