Conn. Gen. Stat. § 33-265: Legal status and powers.

All ecclesiastical societies in this state, in communion with the Protestant Episcopal Church in the United States of America, shall be known in the law as parishes as well as ecclesiastical societies, and shall have power to receive and hold by gift, grant or purchase all property, real or personal, that has been or may be conveyed to them for maintaining religious worship according to the doctrine, discipline and worship of said church, and for the support of the educational and charitable institutions of the same, and shall have and exercise all the ordinary powers of bodies corporate.