Parry adds in the letter, addressed to Culture Secretary Oliver Dowden and seen by The Times newspaper, it is unlikely the top-flight Premier League will be able to fund a rescue package as they have been affected too by the pandemic.
Premier League clubs still, however, managed to splash out £1.2 billion ($1.6 billion) in the most recent transfer window.
"EFL clubs, almost all of them the social cornerstone of the towns and cities they bear the names of, stand on the brink of a financial precipice," Parry wrote.
Parry had recently championed the short-lived Project Big Picture.
It was seen as a power grab from the Premier League's 'big six' of Liverpool, Manchester United, Arsenal, Chelsea, Manchester City and Tottenham Hotspur.
Premier League clubs subsequently offered League One and League Two clubs £50 million which included a £30 million loan.
They rejected it as an act of solidarity with second-tier Championship clubs.
Dowden has previously insisted the 20 Premier League clubs must support the 72 in the EFL, who say they need £250 million to survive a season where fans have been banned from attending games.
However, Parry accuses the government of not treating football on the same level as the arts who were granted a £1.5 billion funding package.
"While football grounds in Rochdale, Grimsby, Mansfield and Carlisle might seem an awful long way from Glyndebourne or the Royal Ballet, they are nonetheless equally important parts of our nation's heritage," wrote Parry in a letter that was also sent to the 72 clubs.
"It must have dawned on you that it is deeply unfair that cultural institutions like these are receiving government hand-outs while also being able to generate revenues by admitting the paying public."
Parry, formerly chief executive of Liverpool, says communities will never forgive the Conservative government if their beloved club goes to the wall.
"Ultimately, the football public will judge the performance of this Conservative government on how many football clubs remain in business once the pandemic finally subsides," says Parry.
"Certainly, those communities that are inextricably linked to their local team will never forgive it if their beloved football clubs are driven into extinction."
Parry, 65, says they are still in talks with the Premier League but claims they may not have the funds to improve their rescue package.
"Discussions with the Premier League continue, however it is clear that top-flight clubs are also feeling the effects of the pandemic, particularly the loss of gate revenue, and it may not be in a position to provide the level of support that is required," he said.
"Therefore, the onus remains on you to remedy the situation rather than thinking of it as 'job done'.
"After all, it is the government that is currently preventing fans from going to games not the Premier League."