In three-to-five-foot barrels, the world number five pushed the limits of high-performance beachbreak surfing to post the day's highest heat total of 18.33.
Parkinson knocked out Jordy Smith as he bounced back from defeat in round four to reach the last eight.
“It was just one of those heats where I waited for a while and got one good one and thought ‘that could be amazing, just that one wave',” Parkinson said.
“Then I got another one. Poor Jordy really got nothing. It was a lot like home, the good ones have a really nice draw on the water off of the bank.
"It’s really dumpy, so you air drop into it and it’s hard to find a rail. Who would have thought you’d come to Brazil and get two nines getting barrelled?"
Parkinson next faces Tiago Pires of Portugal.
Another Australian, Josh Kerr, was also on good form, the 28-year-old notching 17.84 to see off the challenge of defending event champion Adriano de Souza of Brazil, who had scored a perfect 10 early in the heat, and Frenchman Jeremy Flores.
“I saw Adriano’s wave go down and we were sitting in a different spot than everyone else,” Kerr said. “We were getting nines and 10s and I feel bad for Adriano. I wasn’t expecting a little backdoor shootout and I’m stoked to get some good waves.”
De Souza's interest in the competition remains though after he beat Taj Burrow in a fifth-round match-up to set up a quarter-final meeting with Kerr.
"I was focused on the right, but I knew there was a little channel and I went over to the left,” De Souza said. “Luck is really going my way and I’m stoked. The waves were really hard and I’m stoked to make it through this heat.”
QF 1: Tiago Pires (PRT) vs. Joel Parkinson (AUS)
QF 2: Alejo Muniz (BRA) vs. Mick Fanning (AUS)
QF 3: John John Florence (HAW) vs. Julian Wilson (AUS)
QF 4: Josh Kerr (AUS) vs. Adriano de Souza (BRA)
Heat 1: Joel Parkinson (AUS) 18.33 def. Jordy Smith (ZAF) 7.00
Heat 2: Mick Fanning (AUS) 14.00 def. Michel Bourez (PYF) 7.20
Heat 3: Julian Wilson (AUS) 11.43 def. Jeremy Flores (FRA) 10.77
Heat 4:Adriano de Souza (BRA) 12.73 def. Taj Burrow (AUS) 7.90
Heat 1: Tiago Pires (PRT) 14.40, Jordy Smith (ZAF) 12.67, Michel Bourez (PYF) 7.50
Heat 2: Alejo Muniz (BRA) 11.90, Mick Fanning (AUS) 11.43, Joel Parkinson (AUS) 10.73
Heat 3: John John Florence (HAW) 17.73, Julian Wilson (AUS) 17.50, Taj Burrow (AUS) 13.20
Heat 4: Josh Kerr (AUS) 17.84, Adriano de Souza (BRA) 14.10, Jeremy Flores (FRA) 5.97
Heat 3: Jordy Smith (ZAF) 17.37 def. Kolohe Andino (USA) 6.33
Heat 4: Mick Fanning (AUS) 13.00 def. Bede Durbidge (AUS) 4.17
Heat 5: Alejo Muniz (BRA) 13.34 def. Heitor Alves (BRA) 8.50
Heat 6: Joel Parkinson (AUS) 13.07 def. Peterson Crisanto (BRA) 12.67
Heat 7: Taj Burrow (AUS) 14.50 def. Patrick Gudauskas (USA) 13.70
Heat 8: John John Florence (HAW) 16.23 def. Miguel Pupo (BRA) 14.14
Heat 9: Julian Wilson (AUS) 15.90 def. C.J. Hobgood (USA) 8.94
Heat 10: Josh Kerr (AUS) 10.73 def. Adam Melling (AUS) 7.33
Heat 11: Jeremy Flores (FRA) 13.33 def. Adrian Buchan (AUS) 10.17
Heat 12: Adriano de Souza (BRA) 15.10 def. Travis Logie (ZAF) 11.77