The round of 96 was completed and eight heats of the 12 in the round of 48 also finished on two-to-three-foot waves at Merewether Beach in Newcastle, New South Wales.
Australian Burrow battled through to win the latter-round heat over Brazilians Filipe Toledo and Jesse Mendes and American Michael Dunphy.
"I knew it was going to be a tough heat," Burrow said. "Those guys are all energetic in small surf like that. They all rip.
"Last year I felt good here and had an unfortunate interference in the quarter-finals and I lost. That’s always disappointing, but that was my mistake and I’ll never do that again.
"I’m hoping to do better than the quarters this time. My board feels great and I feel great so there’s no reason why I can’t go all the way."
His compatriot Parkinson finished third in a tight heat behind Alan Jhones and Shane Holmes.
Another Australian Jack Freestone won his round-of-96 heat and will contest his next match-up against David Vlug, Nathaniel Curran and Alan Jhones.
"I was looking at the results at which heat I would go into and I saw that I either had Mitch [Crews] or Parko (Parkinson),” Freestone said. “So either way I knew I had a tough heat on my hands.
"That’s the first heat I’ve had with Mitch for a long time and it’s awesome that we both got through. We’re great mates, we travel together and we’re moving into the same house soon too."
The action in Newcastle continues until March 18.