It is not clear to me if both the parents and the judge have to approve, but that is what it appears. It would seem under 16 need both parent and judge. 17-18, just parent.
Here is the actual statute:
Marriage of minors “(a) No license may be issued to any applicant under sixteen years of age, unless the judge of probate for the district in which the minor resides endorses his written consent on the license. (b) No license may be issued to any applicant under eighteen years of age, unless the written consent of a parent or guardian of the person of such minor, signed and acknowledged before a person authorized to take acknowledgments of conveyances under the provisions of section 47-5a, or authorized to take acknowledgments in any other state or country, is filed with the registrar. If no parent or guardian of the person of such minor is a resident of the United States, the written consent of the judge of probate for the district in which the minor resides, endorsed on the license, shall be sufficient.” [Emphasis added]
Marriage in CT.