It was set up by a 32-yard completion to David Tyree with 59 seconds to go on which the receiver trapped the ball against his helmet while being guarded tightly by Rodney Harrison.
After a brief break for bad light, left-armer Parnell struck as Strauss, who had reached 22 despite some hostile treatment from the new-ball pair, was trapped lbw by a ball that nipped back a fraction.