American businessmen Gillett and Hicks took over at Anfield in early 2007 and their tenure lasted until late 2010, when their compatriot John W Henry assumed control.
The 2009-10 season was the last time Liverpool appeared in the Champions League and they have failed to finish higher than sixth in the Barclays Premier League since coming second in 2008-09.
Regarding the Hicks and Gillett era, Parry, who was Reds chief executive at the time of their takeover and departed the post in 2009, told BBC Radio Merseyside: "That period of destabilisation is when we fell out of the Champions League.
"Once you're in there, you're achieving those regular revenue streams. It's a mountain to climb when it falls away."
During the reign of Gillett and Hicks - whose relationship was described by the former in 2008 as having been "unworkable for some time" - the debt against Liverpool's holding company, stemming from a Royal Bank of Scotland loan the pair had used to help them purchase the club, continued to grow.
"The debt was never on the club because previous chairman David Moores and I blocked that," Parry said.
"It was in the holding company, but clearly they needed to get money from the club to pay the interest so that was going to be a burden.
"Our clear understanding was that it was short-term debt while they reorganised their own finances.
"It was in the offer document that there would be no debt on the club, or the club would not have to bear the costs, and certainly that changed."
He added: "That (2008-09) is maybe the best ever chance we've had to win the Premier League and that was a time where we should have been kicking on.
"With a few more adjustments to the squad, who knows what might have happened in the next year or two.
"You'd do a million things differently with hindsight, but that's not an option. We all wish passionately that they hadn't owned the club."