Schmelzer was given his marching orders for handball, with Diego sweeping in the spot-kick to make it 1-1.
And Stark admitted after the game it had been an error.
"I am sorry, but it was an error in interpretation from me," he said. "We have taken a look at it again in the dressing room, together with the referee observer.
"During the game, to me it seemed a clear handball on the goal line, so penalty and red card.
"Unfortunately, the pictures speak a different language. It shouldn't happen, but it has happened.
"The penalty and the red card were mistakes."
The admission does not help Dortmund get over their second home defeat of the season, though, as their deficit on league leaders Bayern Munich grew to 14 points.
"We have a case here where Stark has completely decided this game against us with his refereeing and that is obviously hard to accept," said the club's director of sport Michael Zorc to Sky television.
Coach Jurgen Klopp added that Schmelzer "should receive the peace prize" for his reaction to the injustice as he left the field with little complaint, but he also felt it was a mistake which should never have been made.
"I was certain it wasn't handball from where I was standing - his arms were attached to his body," Klopp said.
"We're not at the point where players have to saw off their arms to avoid handball, yet.
"The way this result came about is brutally hard for us to accept."