Cherries chairman Eddie Mitchell released a statement announcing the departures less than 24 hours after the 3-1 defeat to Crawley which sent Bournemouth fifth-from-bottom in League One.
In his statement, Mitchell praised the duo for their hard work since taking over in May.
He said: "I would urge our supporters to recognise the time and effort that Paul and Shaun invested in the first team and their unwavering determination to produce an attractive brand of football.
"As chairman of AFC Bournemouth, I have personally recognised the direction in which they were trying to take the team in terms of tactics and style of play, and throughout their time in charge they have had my full support.
"However, I believe that luck has not been on their side during their short period at the helm. As we know, football is a results business, but most people will be unaware of the hard work and dedication that they put in behind the scenes to help this club move forward."
He added that both Groves and Brooks personally asked him to thank the fans for the support they gave the management team during their time in charge.