Parker picked up a minor knee injury in training on Wednesday and is due to undergo medical tests today and in the run-up to kick-off.
Coach Joel Santana reported: "Bernard had a slight problem and he was advised not to continue training.
"He'll be having check-ups to see if he is able to play. It'll be a shame if he doesn't take part tomorrow as he's a new and spontaneous talent."
The Red Star Belgrade forward has been instrumental in getting South Africa into the last four, with his two goals sinking New Zealand to help the hosts take the runners-up spot in their group.
That brace was sweet redemption for Parker after he unwittingly blocked Kagisho Dikgacoi's goal-bound header in the 84th minute of the opening 0-0 draw with Iraq.
The winner of Thursday's game will play the victor of Wednesday's semi-final in Bloemfontain between Spain and the USA.