A proposed "new deal" for English football would see third and fourth-round FA Cup replays and two-legged League Cup semi-finals abolished to help make more room in the calendar for UEFA's expanded European competitions from next season.
In return, top-flight clubs were expected to sign off on an extra £900 million of funding for the pyramid over the next six years.
However, the deal is yet to be ratified with Premier League clubs believed to be divided on the level of extra funding and who should bear the brunt of that extra cost.
FA Cup replays can be worth hundreds of thousands of pounds to lower league clubs through gate receipts and television revenue.
"The Premier League's position has been (that) the rest of the game has to change and move to accommodate them," said Parry, as he addressed a Culture, Media and Sport committee meeting on Tuesday.
"We did suggest remedy was within the Premier League's hands, they could reduce to 18 clubs (from 20), that would solve the problem - that didn't seem to go down very well.
"As part of a new deal, we're prepared to take on board considerable loss of revenue from the loss of FA Cup replays (and) the second legs of the Carabao (League Cup). We're absolutely not prepared to concede those on the basis there is no deal."
A number of Premier League managers have criticised the continuation of FA Cup replays in recent weeks.
Half of the top-flight's 20 clubs did not receive a planned two-week winter break due to either being involved in third-round replays this week or the League Cup semi-finals in January.
Replays from the fifth round and on have already been scrapped.
"A replay, I don't get it," said Brentford boss Thomas Frank, whose side face Wolves in a FA Cup replay on Tuesday.
"I think tradition is important, we need to protect it in a way. But it's already been cut away a bit. We only have replays in the third and fourth round. It's either all or nothing.
"Cut it. Protect the best cup tournament in the world and make it even more important. Take the replays out and put even more focus on this fantastic tournament."
