The group's letter to Oliver Dowden, the Secretary of State for Digital, Culture, Media and Sport, has emerged two days after the family of World Cup winner Nobby Stiles said they had been told by experts his brain was severely damaged by repeatedly heading the ball.
In the letter, the group referenced the publication last year of the FIELD study, which sought to determine if there was a link between heading the ball and an increase risk of diseases such as dementia.
It said: "While the report said there was no definitive link, it confirmed that players were three and a half times more likely to die of dementia and other neurological diseases than the general population.
"Following the study's publication, the FA in parallel with UEFA's medical committee published new guidelines that apply to all young players.
"Measures included a complete prohibition of header training for children below the age of 12 and a graduated process to headers thereafter.
"We welcome these measures but believe they do not go far enough."
The letter concluded by urging Dowden to ask Parliament "to review the links between neurological diseases and heading the ball and whether the guidelines are adequate to protect players at every level of the game".
The letter is co-ordinated by former Charlton and Chelsea player Mickey Ambrose and, besides Reid and Anderson, it is also signed by Clive Wilson, Dean Wilkins, Vinny Samways, Mark McCammon, Darren Moore and Phil Brown.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies