The Daily Mail says the futures of both men remain in doubt after the two held a brief meeting prior to Tuesday's Hillsborough memorial service.
Benitez was furious when co-owner Tom Hicks revealed Parry was present at a meeting with Jurgen Klinsmann, a possible replacement for the Liverpool manager. Hicks has demanded the resignation of Parry but George Gillett, Hicks' feuding business partner, has backed the chief executive.
Benitez and Parry discussed the situation in private before the Anfield service on the 19th anniversary of the Hillsborough tragedy, which cost 96 Liverpool fans their lives. They sat a few seats apart at the service. Neither man commented.
Benitez, who led the first-team squad at the service, said: "This is obviously an important day of the year for everybody connected with our club. We must take time to remember those who lost their lives at Hillsborough. We must give them our support and show that we are thinking of them."