Media reports have indicated the former England international is heading for Championship (second tier) side Bournemouth as a replacement for Jonathan Woodgate.
Parker was appointed caretaker manager in February 2019 with the club heading for relegation but brought them back up to the top flight last August for what proved to be a short-lived return.
"Through promotion and relegation alike, Scott has always enjoyed my support as our head coach," said chairman Shahid Khan in a statement on the club website.
"Scott’s departure does nothing to shake my confidence, however. We will hire a new head coach who is capable of achieving our goal of promotion and will be committed to Fulham and its supporters."
Fulham said first-team coach Matt Wells and goalkeeping coach Rob Burch were among a number of backroom staff who had also left the club.