John and Beverley Taylor, both 66, were discovered inside 3 Regent Crescent on December 21.
David Michael Taylor, 35, who is believed to be their son, has been charged with their murders but was unfit to be produced at his first appearance at Leeds Crown Court shortly before Christmas. He will undergo further psychiatric assessments in custody at HMP Hull.
A statement released by North Yorkshire Police said: "The family of John and Beverley Taylor (née Bernard) would ask through this devastating time they be allowed their privacy. They would like to thank all the friends and the community of Skipton for all their support and messages of comfort. Thank you."
Taylor will next appear in court on January 7 but his case is likely to be transferred to Bradford Crown Court.