Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts.
Liam Rosenior has thanked the fans of Hull City for "five wonderful years".
Goalkeeper Jayson Leutwiler, defenders Lewis Sirrell and Birger Meling, plus midfielders Jake Fowler, Cameron Park and Matthew Waters will all look for new clubs this summer.
But the club have offered deals to defenders Jonathan Burn and Kieran Weledji, midfielder Mark Kitching and striker James Warnett.
Parnaby, who returned for a second spell at the club in July 2012, made a total of 158 appearances, while former Leeds full-back Richardson made just one appearance under Karanka before spending the last three months on loan at Ipswich.
Nathaniel Chalobah, Danny Graham, Tomas Mejias, Kenneth Omeruo and Jozsef Varga, all on loan at the club, have returned to their parent clubs.