Peterson, 28, was among the early starters who got through most of their rounds before wet weather left waterlogged parts of the two courses used for the tournament.
Peterson, who won his first European Tour title at the Czech Masters last year, hit nine birdies in a nine under par 62 on the West Course.
Aaron Rai and Paul Waring were joint second after carding seven under 65 on the tougher East Course.
Just under half the field had still to complete their rounds when play was called off for the day. It will resume early on Friday, when more wet weather is forecast.
The field of 210 players is the biggest of the year on the European Tour. Each player plays one round on both courses at the club over the first two rounds.
Three players from the top 10 who are not already exempt for the British Open will win a place in the tournament, to be held at Royal Birkdale from 20-23 July.
(Reporting by Mark Gleeson in Cape Town; Editing by Larry King)
