The Clarets confirmed Parker had left the club by mutual consent on Thursday, just 8 days after the club's relegation from the Premier League was confirmed.
Parker, 45, won promotion with Burnley last season, but his side failed to survive in the top flights after a number of very poor performances sent them straight back down to the Championship.
Now, as the club mull over options such as former Rangers and Aston Villa boss Steven Gerrard, the BBC report that Burnley are seeking to appoint Bellamy.
Bellamy has been in charge of Wales for two years, but they missed out on the World Cup after losing in the play-offs to Bosnia and Herzegovina in March.
The former midfielder only took charge of Wales in July 2024 on a four-year deal and despite missing out on the World Cup, will be looking ahead to the UEFA Euro 2028, which will be co-hosted across the UK and Ireland.
The BBC report that Bellamy was among the candidates to take the top job before Parker's appointment in 2024 and that Burnley view him as a serious candidate for the position.
Bellamy and Gerrard are both favourites for the role ahead of the end of the season where Burnley will make the decision heading into the Championship campaign as one of the favourite teams to go straight back up.