Cherries co-owner Maxim Demin said: "I would like to place on record my gratitude to Scott and his team for their efforts during their time with us.
"Our promotion back to the Premier League last season under his tenure will always be remembered as one of the most successful seasons in our history.
"However, in order for us to keep progressing as a team and a club as a whole, it is unconditional that we are aligned in our strategy to run the club sustainably.
"We must also show belief in and respect for one another. That is the approach that has brought this club so much success in recent history, and one that we will not veer from now. Our search for a new head coach will begin immediately."
Gary O'Neil will take interim charge of the first team, assisted by Shaun Cooper and Tommy Elphick.
Parker arrived at the Vitality Stadium on June 28, 2021 - the same day his departure from Fulham was announced.
The 41-year-old took over from Jonathan Woodgate and was handed a three-year-contract.
Parker got off to a flyer in the Championship and picked up the Manager of the Month award for September 2021 after winning 13 of a possible 15 points.
The club finished the season in second and earned automatic promotion back to the Premier League.
Parker enjoyed another solid start in the top flight as his men picked up a 2-0 victory over Aston Villa.
However, Bournemouth lost their following three games by a combined scoreline of 16-0, with the run culminating at Anfield where the Cherries were handed a 9-0 hammering by Liverpool.