Defeat against Uruguay in Sao Paulo left the Three Lions on the brink of an ignominious exit from the competition, and Costa Rica's shock defeat of Italy confirmed their fate.
Parker, who played in England's 1990 semi-final defeat to West Germany and set up Gary Lineker's equaliser in the famous Turin game, claims that Hodgson is the wrong person to take the national team forward at a crucial time in the development of young talent.
He told Eurosport: "We need a new manager in who is going to inject some dynamism into the squad, who can be trusted to bring the youngsters on. I don't think Roy is the man to do it for England moving forward. This tournament has not been kind to his, or his team's capabilities.
"When you sit and watch Costa Rica outplay an Italy side who have nothing going forward, you realise just how bad England's result was last Saturday in losing 2-1. This Italian side struggle to create anything when you apply some pressure. You have to ask yourself why England were not applying any pressure last weekend?
"This is the quickest end to a tournament in England's history. It is a complete farce."
England last exited a World Cup at this stage in 1958.