The former England midfielder helped the Cherries return to the Premier League as Championship runners-up last season but endured a challenging return to the top flight.
With concerns over their summer recruitment and several poor displays in the opening weeks of the season, Parker was sacked just days after his side crashed to a 9-0 defeat at Liverpool.
Cherries owner Maxim Denim spoke of the decision in a club statement, saying: “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.”
Parker’s search for a new role has taken a surprising twist with reports in France suggesting he could be handed a chance to take over at Ligue 1 club Nice.
French outlet FootMercato have claimed Les Aiglons are considering replacing current manager Lucien Favre after his side collected just two wins from their opening eight games.
An approach was reportedly made to former Tottenham Hotspur and Paris Saint-Germain manager Mauricio Pochettino - but the Argentinian is keen to bide his time for a more high-profile role.
Parker is said to be Nice’s managerial shortlist and could be approached if they decide to part company with Favre.
The ex-Magpies midfielder will find a number of familiar faces if he takes over at the Allianz Riviera with former Premier League stars Ross Barkley, Kasper Schmeichel, Morgan Schneiderlin, Joe Bryan, Aaron Ramsey, Nicolas Pepe and Mario Lemina all part of the current Nice squad.