Parejo became the fourth player in team history to complete the feat, but Lancaster fell to the visiting Visalia Rawhide, 10-9, in a roller-coaster game.
Mike Baxter cycled for Triple-A Portland earlier in the day in the Beavers' 13-8 win over the Reno Aces.
Parejo homered for the second time this season in the second inning. He singled in the fourth, plated two runs with his second triple in the eighth ahead of the ninth-inning two-bagger.
"I feel good today," he said. "I hit the ball very well."
Originally drafted by the Milwaukee Brewers in 2003, the outfielder worked his way as far up the ladder as Triple-A Nashville last season before signing on with Houston as a Minor League free agent Dec. 2.
Parejo has fared very well with the JetHawks this season, batting .318 with 25 RBIs and 25 runs scored. He had two homers and one triple through 108 games last year between Double-A Huntsville and Nashville.
He was the first Lancaster player to cycle since Andy Green did so at High Desert on Aug. 21, 2002. Pee Wee Lopez (Aug. 15, 1999 against Lake Elsinore) and Juan Silvestri (May 3, 2000 vs. Bakersfield) also accomplished the feat.
Reliever Patrick Urckfitz made the first start of his professional career for the JetHawks, allowing two runs -- one earned -- on five hits without a walk and four strikeouts in five innings, but didn't figure into the decision.
"He never started before," Parejo said. "For me, he did very well."
But the bullpen ran into difficulty, with Leandro Cespedes tagged for three runs on three hits in two-thirds of an inning, Ashton Mowdy allowing four unearned runs on a hit and two walks in two frames and Kyle Godfrey (2-2) suffering the loss after giving up three runs -- one earned -- on four hits in 1 1/3 innings.
"We played hard today," Parejo said. "We tried."