Co-owner Hicks demanded Parry's resignation as chief executive in a letter sent to Anfield yesterday. But he is unable to force Parry out as the American only controls 50 per cent of the club, leaving him powerless to wield the axe.
The Sun says the Texan billionaire is believed to hold Parry responsible for Liverpool's poor commercial performance.
Their £133.9m turnover is nearly £80m adrift of Manchester United.
Hicks also claims Rafa Benitez has told him he is unhappy at the way Parry has handled transfer dealings - and the Kop boss has had an uneasy relationship with the chief executive for some time.
Yet rather than casting a shadow over the Spaniard's future as well, it actually places Benitez in a stronger position with Hicks throwing his support behind the manager.