The former Fulham boss guided the club to promotion from the Championship last season however they have lost three of their opening four Premier League games without scoring, including a 9-0 drubbing at the hands of Liverpool.
In Tuesday’s announcement, Cherries owner Maxim Demin stressed the need for “respect” between himself and the manager and hinted at a disagreement with Parker over running the club “sustainably”.
He 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.”
Bournemouth have only paid fees for two first-team signings since winning the promotion, alongside a number of free transfers.
Gary O’Neill will take temporary charge of the visit of Wolves on Wednesday.