The Aston Villa midfielder has been poised for a switch to Anfield all summer but the Reds have been unwilling to match the asking price of £18million (€23m).
Last month, Villa set Liverpool a deadline to submit an acceptable offer but none was forthcoming, and Villa then announced the deal was off.
However, a few days later manager Martin O’Neill conceded Barry’s heart was still set on the move, leaving the way open for Liverpool.
But the expected transfer has still not happened, amid speculation Liverpool’s American owners are not willing to fork out the money following Robbie Keane’s £20.3m (€26m) move from Tottenham.
In a statement, Parry insisted George Gillett and Tom Hicks are backing Liverpool boss Rafa Benitez.
“There has been a lot of comment regarding our position with regards to Gareth Barry,” Parry said.
“This is not about questioning the manager’s judgment or the ability of a particular player. The owners have clearly demonstrated throughout the year they are willing to back Rafa in the transfer market and will continue to do so.
“It is obviously the selling club’s prerogative to put whatever price they want on the player, but on this occasion Liverpool think the price quoted is too high.”
Meanwhile Villa midfielder Steve Sidwell is hopeful that Barry will remain at Villa Park. Sidwell, who joined O’Neill’s side from Chelsea this summer, wants his fellow midfielder to stay put as the club look to break into the top four.
Sidwell said: “You don’t want your best players to leave, especially with the club where we are at the moment.
“We want to be pushing on and to do that you need to keep all your best players and bring players in. If he stays it is fantastic, but if it does happen that he does go, we’ll wish him all the best. But hopefully he will stay.”