Following the sacking of Kenny Dalglish, the former chief executive says that qualifying for Europe's biggest club tournament should be the first priority for any new manager
Parry, who left the club in 2009, feels that, despite the fact that such an aim is now more difficult than before, the Anfield club should have at least been in contention for the top four in 2011-12.
"We always said to be in the Champions League isn't a goal, it's a minimum requirement," he told BBC Sport.
"It's tougher now but I don't think it was unrealistic to at least be looking at being in contention last year."
|MARTINEZ IN THE FRAME
|Wigan manager set for second interview with FSG over vacant Liverpool role
He explained: "If you look at the criteria we applied when we appointed Gerard and Rafa they were a little different, even in that five-year period.
"With Gerard we deliberately decided we wanted to bring in some continental experience. With the advent of the Champions League, France were very much in the ascendancy, so we thought he would bring something fresh.
"We parted company with Gerard having qualified for the Champions League because our standard then was we wanted to be in contention for winning the league.
"We wanted someone young and hungry, not someone looking for a pension. Rafa, having won the league twice with Valencia, was a perfect fit."
As Liverpool are said to be considering younger managers such as Roberto Martinez and Andre Villas-Boas following the sacking of Kenny Dalglish, their past experience and current market rate are sure to be high on the list of requirements, as they were during Parry's tenure.
"You first of all set the criteria very tightly," he noted. "From my experience that led us to a shortlist of one. In each instance we got the person we were looking for.
"In every negotiation there is skill, a bit of luck and the third element is timing. Sometimes that is in your favour, sometimes it isn't because it is all about the right people being available."