9 September 2013 Last updated at 20:23 BST
The Independent Police Complaints Commission has said Det Supt Steve Fulcher, from Wiltshire Police, should face disciplinary charges over the investigation.
It follows a judge's ruling that Mr Fulcher, ignored guidelines which meant no-one has been prosecuted for Miss Godden's death.
Christopher Halliwell led him to Miss Godden's buried body after admitting Sian O'Callaghan's murder in 2011.
Miss Godden's father John Godden said he found the critical police watchdog report "shocking".
However, her mother Karen Edwards said she supported Det Supt Fulcher and said she thought he "deserved a medal".
The parents of murdered Becky Godden have spoken about their differing views on whether a detective in the case should face disciplinary charges.