The Spaniard knew about the meeting with the former Germany striker but was surprised to hear of Parry's involvement.
Klinsmann has since agreed to take over at Bayern Munich in July.
"I need to talk with the board... as soon as possible," Benitez told reporters after Sunday's 3-1 win over Blackburn Rovers.
"I want to clarify everything.
|"I need to talk with the board... I want to clarify everything."|
"I was surprised with some meetings with another manager. I was surprised with the people that were in the meeting.
"I feel my position is being undermined by it. I need to speak to all of the people. They were in a meeting together so I need to know something more."
The latest outbreak of unrest in a season of behind the scenes turmoil at Liverpool came after Hicks criticised Parry and said he would ask co-owner George Gillett to back his plans to fire the chief executive.
However the call is likely to fall on deaf ears as last month Gillett said his relationship with fellow American Hicks had broken down.
The two co-owners are at loggerheads after takeover talks with Dubai International Capital (DIC) broke down earlier this year.
Hicks accused Parry of arrogance over the weekend and repeated calls for him to leave Anfield.
"Rick Parry has been at Liverpool for 10 years and we haven't won a league championship under his leadership," Hicks told the Mail on Sunday.
"After watching him operate I came to the conclusion it was time to ask him to resign, due to his inability to manage an organisation, his seemingly arrogant attitude to our supporters and his lack of communication with Rafa."
Liverpool are fourth in the English Premier League, 14 points behind leaders Manchester United and out of title contention.
The Anfield club face Chelsea in the Uefa Champions League semi-finals later this month.