The Kaizer Chiefs star looked to have kept his nation's hopes of qualifying for the World Cup in Brazil alive when he gave South Africa a first-half lead but, ended the tie as the villain after heading into his own net in remarkable fashion from an Ethiopian setpiece late on in the encounter.
Ironically, it is from set-pieces that the 35-year-old believes Hugo Broos' side will be at their most dangerous this time around, and he urged the current crop to make most of any dead-ball situations that arise during the match.
"They have speedy players [but] we have the skill to be able to counter them," Parker told Kaizer Chiefs' website. "We are dangerous from set-piece situations and when we get them, we need to make them count for us.
"We need to target to get six points from these games. I think if we can come out with four points, it will be good because then all the pressure is on Zimbabwe and Ghana going into the last two games."
