The former Queens Park Rangers chief has been out of work since leaving Leicester last year and he has been keen to get back into management.
Parkes was understood to be unhappy with the terms offered to him and his backroom staff, and has doubts about the club's plans for next season.
Parkes stepped in to manage affairs after Simon Grayson controversially resigned to manage Leeds. Oyston reported Leeds to the footballing authorites and made a number of serious allegations against United. Following a mutual settlement prior to a Football League hearing a gagging order on the agreement meant the allegations were never dispelled.
Parkes was left with the fallout of Graysons hurried departure and made a number of astute loan signings two of which were of significant impact to steer the club away from relegation and end the season 10 points clear in 16th position.