The Australian duo is knotted at 9-under-par 135 and are one ahead at Kooyonga Golf Club.
Peter Senior (68), Greg Chalmers (69) and American Brad Ott (70) are tied for third place at minus-8.
Parry got off to a great start on Friday with a pair of birdies at his first two holes. He parred his next four, but birdied the seventh and ninth to make the turn at minus-5 for the championship.
Parry started his second nine in familiar fashion as he birdied the 11th and 12th holes. He birdied the par-3 15th, then added another at the last to join Rumford atop the leaderboard.
'The course really laid down this afternoon,' said Parry. 'It was very pleasant out there. I think the guys this morning really got the tough end of the draw.'
Parry is perhaps the most accomplished player in this field. He owns two PGA TOUR wins - the 2002 WGC-NEC Invitational and the 2004 Ford Championship at Doral.
He has 20 international wins, but does not care to play the PGA TOUR anymore. Parry wanted to stay closer to home and compete on the Australasian circuit.
Rumford, like Parry, does not call the Nationwide Tour his home either. He is a member of the European Tour, but elected to play this event in his home country.
He tallied his first birdie of round two at the par-5 second hole, then recorded his second birdie at the fifth. Rumford dropped a shot at the eighth hole, but atoned for the mistake with a birdie at No. 9.
Rumford parred his next eight holes, but knocked his approach inside 3 feet at the last. He converted the short birdie putt to match Parry's 9-under-par total.
'Probably played a little bit better than I did yesterday but I guess it was masked by the fact that I just didn't putt as well today,' said Rumford.
Steven Jeffress (70), Andrew Pitt (70), Roger Tambellini (68), Scott Gardiner (69) and Anthony Brown (70) are knotted in sixth place at 7-under-par 137.
Overnight leader David Lutterus struggled on Friday. He shot a 3-over-par 75 and is tied for 17th place at 5-under-par 139.
The 36-hole cut fell at 2-under-par 142 with 63 players advancing to the weekend.