Parker's future at Burnley is being discussed following the club's relegation from the Premier League.
The 45-year-old led the Clarets to promotion last season but has been unable to keep the club in the top flight. It's a third career relegation for Parker as a manager, following two previous relegations with Fulham.
Despite his impressive record of second-tier promotion, leading Fulham, Bournemouth and Burnley into the Premier League, the Burnley board are considering change. No decision has been made but a mutual exit is possible.
According to Ben Jacobs, Gerrard and Bellamy are potential successors. Wales manager Bellamy was previously assistant manager to Vincent Kompany at Burnley, before taking charge of the Welsh national team.
The 46-year-old has the third-highest win percentage in Wales history (44%) but was unable to lead The Dragons to the 2026 World Cup. Wales were beaten on penalties by Bosnia and Herzegovina on penalties in the play-offs.
Steven Gerrard is another option for Burnley and is open to a return to management. The ex-Rangers manager has been without a club since leaving Saudi side Al-Ettifaq.
The 45-year-old led Rangers to the Scottish Premiership in 2021, without losing a game, before stints with Aston Villa and in Saudi Arabia. He is understood to be open to a Championship role.
Earlier this month, Gerrard told The Overlap that he was unsure on what his future held. Asked about the possibility of not going back into management, he said: “I’m okay with that, I enjoy covering some of the games.
"I enjoy having quite a bit of free time, probably the best part I’ve had from a family point of view in ever really, in terms of being there, being present.
"Lio [Gerrard's youngest son] is eight now. During his career, I’ve been in Scotland, in Birmingham, so yeah, really enjoying the family side of it and having a load of free time, but also I’m conscious that I want to stay busy enough.”