The EFL has been calling for the Premier League to share 25% of its media revenues, but talks collapsed in March. Parachute payments are among the reasons the two leagues have been unable to reach a new financial settlement worth an average £125m per year extra, despite years of negotiations and pressure from politicians.
The EFL believes the payments - worth tens of millions of pounds to relegated clubs - distort competition and encourage reckless spending among Championship clubs. The Premier League insists the payments are essential to give club owners the confidence to invest.
After they were excluded from the previous bill, Parry is "pleased" that the regulator can now consider them if deemed of risk to financial sustainability.
"We’re not saying they should be abolished," said Parry.
"They should be properly considered alongside other distributions, so the fact they are is good. Because they’re a proportion of the Premier League’s TV revenues, they have exploded over the last 20 years.
"They’ve just got way too big and it needs a rethink. It really is about how do we make sure relegated clubs are protected, but there’s fair competition within the Championship.
"We’re all about sustainability and to achieve that what we really need is the position where we have sporting jeopardy without financial catastrophe."