The duo are among four members of the backroom team, along with development coach John Dungworth and kit man Kurt McDermott, to depart as part of a reshuffle behind the scenes at the Proact Stadium.
Wright was named as former manager John Sheridan's assistant in June 2009 after the pair previously worked together at Oldham.
After helping the Spireites win the League Two title and the Johnstone's Paint Trophy, he had a spell as caretaker manager this season following Sheridan's departure.
Crossley also followed Sheridan from Oldham in the summer of 2009. The former goalkeeper assisted Wright during his time in temporary charge and looked after the reserves with Dungworth.
Chesterfield boss Paul Cook said: "On behalf of the club, I'd like to thank all the staff for their efforts and the support they gave me. I wish them all the best for the future."