The Australian, currently ranked second by the ASP, registered the highest single wave score - a 9.93 - in his third round clash before following that up with a heat total of 17.87 to win his fourth round heat.
“That wave felt good,” said Parkinson, who enjoyed the building four-to-six foot glassy waves on offer.
“It was a perfect little Lowers wall. That first real good hit was sweet, it allowed me to feel confident from the first turn. I feed off of Taylor out there. He’s such an amazing veteran and experienced. He’s one of those guys that everyone is cheering for. He deserves a few wins this year and I’d be stoked to see him get a result.”
Parkinson's form sees him skip round five and advance directly to the quarter-finals.
Also heading straight through is John John Florence, the world number three from Hawaii, who claimed the day's highest heat total - an 18.57 - in his round four encounter with Jordy Smith and Adriano de Souza.
“It felt good to get some good scores after that slow heat with Yadin (Nicol),” Florence said. “The waves looked fun and I don’t know if it was me or the waves, but that heat ended up being weird.
"I switched my board up and went out with a fresh start and ended up having a good time in Round 4. I’m so stoked to be straight in to the quarters because those round five heats tend to be the gnarliest heats of the event.”
Defending champion Kelly Slater, seeking an historic 50th ASP World Tour win, posted an easy victory in round three but could only finish third in round four, meaning a fifth round meeting with Jeremy Flores.
“My back is against the wall to be honest,” Slater said. “Every heat is really important if I want to be in the title race at the end of the year. I don’t really have any room to move after missing Brazil and just a stupid mistake in Tahiti where Ricardo got me.
"Those are my two throwaways for the year, but sometimes that’s good though. It helps you keep your focus. I’ve just got to make the right choices and hope I don’t get smashed by someone early on.”
Round five match-ups:
Heat 1: Julian Wilson (AUS) vs. Mick Fanning (AUS)
Heat 2: Jeremy Flores (FRA) vs. Kelly Slater (USA)
Heat 3: Josh Kerr (AUS) vs. Jordy Smith (ZAF)
Heat 4: Adriano de Souza (BRA) vs. Gabriel Medina (BRA)
Round four results:
Heat 1: Taj Burrow (AUS) 12.93, Julian Wilson (AUS) 12.60, Kelly Slater (USA) 11.20
Heat 2: Adrian Buchan (AUS) 17.23, Jeremy Flores (FRA) 15.97, Mick Fanning (AUS) 10.17
Heat 3: Joel Parkinson (AUS) 17.87, Gabriel Medina (BRA) 15.40, Josh Kerr (AUS) 13.63
Heat 4: John John Florence (HAW) 18.57, Adriano de Souza (BRA) 16.27, Jordy Smith (ZAF) 12.67
Round three results:
Heat 1: Kelly Slater (USA) 15.70 def. Matt Wilkinson (AUS) 10.00
Heat 2: Julian Wilson (AUS) 14.03 def. Alejo Muniz (BRA) 12.34
Heat 3: Taj Burrow (AUS) 15.60 def. Adam Melling (AUS) 14.30
Heat 4: Jeremy Flores (FRA) 16.20 def. Kai Otton (AUS) 12.23
Heat 5: Adrian Buchan (AUS) 18.13 def. Heitor Alves (BRA) 8.93
Heat 6: Mick Fanning (AUS) 15.34 def. Dusty Payne (HAW) 13.33
Heat 7: Joel Parkinson (AUS) 18.33 def. Taylor Knox (USA) 13.33
Heat 8: Gabriel Medina (BRA) 17.37 def. Michel Bourez (PYF) 17.06
Heat 9: Josh Kerr (AUS) 13.53 def. Bede Durbidge (AUS) 9.67
Heat 10: Adriano de Souza (BRA) 17.20 def. Travis Logie (ZAF) 15.77
Heat 11: Jordy Smith (ZAF) 14.67 def. Brett Simpson (USA) 10.90
Heat 12: John John Florence (HAW) 14.44 def. Yadin Nicol (AUS) 9.90