It was the first time since April 25-27, 2003, that the A's have swept the Indians in a three-game series, although the A's did sweep a four-game set against Cleveland in 2009.
Crisp had three hits on the day, matching his total from the previous six games. He also improved his batting average with runners in scoring position to .356 (21-for-59).
"He's a guy who we rely on at the top of the order," A's manager Bob Melvin said. "He creates havoc up there, and he also comes up big."
Brandon Moss hit a home run and Jemile Weeks doubled home a run for the A's, who have won four straight.
"It's nice to have guys like Weeks at the bottom of the order, who can get on base for Coco, who constantly comes through up so big in big situations," Melvin said. "To win three like this gets you past a difficult period."
Parker (8-7) needed only 93 pitches to cruise through eight innings. The rookie right-hander, who had allowed 18 runs over his previous 23 innings, struck out two and walked one.
"Our starters are doing a fantastic job," Crisp said. "[Parker] was staying around the strike zone and he has amazing stuff."
After losing his last three starts, Parker said he was able to locate his pitches.
"I had to get out of some jams, and I was able to keep the same rhythm," Parker said. "I was able to throw a lot of two-seamers down in the zone and kept the ball on the ground."
Melvin thought a game like this would wipe out anything negative that's happened in Parker's previous starts.
"He was outstanding," Melvin said. "After a rough start, he was on cruise control. His ball was coming out of his hand well, and you know the potentially poor period he was having is over now."
Despite Parker having thrown fewer than 100 pitches, Melvin decided not to send the rookie to the mound for the ninth, which was handled with ease by Ryan Cook.
"I wanted him to get the complete-game shutout, but I didn't think it was worth it in the end. He's getting up there in innings and you never know how that last inning might have played out."
Weeks, Moss and Chris Carter each had two hits for the A's, who are 22-12 since the All-Star break, the best mark in the American League.