In recent weeks, Parker interrupted Achilles injury. He also predicted would not be able to join the England squad at their training camp in Spain starting on May 21.
"He had problems with his Achilles. I hope it will not deteriorate. This is a long-term problems," said Spurs manager, Harry Redknapp, who told Soccernet.
"He is struggling for fitness. This is bothering him.
After Sunday he would be shot and we'll see if it could heal," he added.
"He already had it in weeks. Perhaps the only cure is a little break, but maybe the injection will cure it. Specialist thinks it could be cured, but will need to rest 10 days after that," said Redknapp.
Before the European Cup, England will berujicoba by Norway on 26 May.
Three days later, the Three Lions squad list must be submitted to UEFA. They will then try Belgium at Wembley on June 2 before plunging into the tournament.
England in Group D at Euro 2012. Roy Hodgson made the team will compete with France, Ukraine, and Sweden.