The 45-year-old Englishman took charge of "The Clarets" before the start of last season, following Vincent Kompany's departure. He guided the club back to the top flight at the first attempt, notably achieving a 31-game unbeaten run.
However, the newly promoted side struggled in the Premier League, with their relegation to the Championship confirmed by last week's 1-0 home defeat to Manchester City.
Following the loss, the club's management initially discussed Parker continuing, but ultimately, both parties agreed to part ways. Parker bid farewell to fans via a statement on the official club website. "It has been an honour to manage this club. I've enjoyed every moment of our journey, but now is the right time to move in a different direction. I will proudly remember our promotion to the Premier League, and I wish the fans and the entire club nothing but the best."
The 37-year-old Dúbravka signed a one-year contract with Burnley before the season and has not yet officially decided on his future.