Holy Name of Jesus parish appears to still be active; it is included on the Bridgeport Diocese list of Stamford parishes. Some dioceses have made copies of parish records & these records would be held by the archivist for the diocese. Try contacting the archivist via form on the website www.bridgeportdiocese.com
(click on About Our Diocese dropdown menu & then click on Staff Directory & then click "Contact Us" in the narrative above the telephone numbers). Or you can try calling the Archives Office 203-416-1354.
Most parishes no longer have full-time office staff, so if you want to contact the parish directly, correspondence could be sent to Holy Name of Jesus, 4 Pulaski Street, Stamford 06902-6822. The parish does not have a website.
Hope that this helps.