Brandon Moss also homered and Cliff Pennington scored three times. The A's won their fourth in a row and moved 10 games over .500, matching their high mark of the season.
Shin-Soo Choo singled twice for the Indians, who have lost five straight on their nine-game road trip.
Parker (8-7), possibly pitching to keep his spot in the rotation following three consecutive losses, gave up six hits, struck out two and walked one. He retired 16 of his final 18 batters.
Oakland scored in four consecutive innings against starter Justin Masterson (9-11).
Crisp and the hitters provided the biggest spark but Parker's outing was equally important. The right-hander hadn't won since going eight innings to beat the Yankees on July 21.
Parker (8-7) didn't allow a run despite pitching with runners in scoring position most of the afternoon. The Indians put two runners on in the second, third and sixth but Parker worked out of trouble each time.
That's significant because two other Oakland starters, Brett Anderson and A.J. Griffin, are set to come off the disabled list soon to rejoin the rotation.
Melvin hasn't said which of the A's current starters will be pushed out, though it's reasonable to assume Parker was and still may among those being considered.
Crisp had RBI singles in the third and sixth, sandwiched around his three-run homer off Masterson in the fifth.
Cleveland was shut out for the sixth time this season. It was so bad for the Indians that Casey Kotchman got thrown out rounding first base after his high popup fell between a pair of Oakland infielders for a single.
Masterson, who had a 1.38 ERA over his previous two starts, allowed nine hits and struck out five while losing to the A's for the second time this season.
The Indians haven't had a winning road trip since taking two of three from the White Sox in early May. They fell to 1-5 on their current trip heading into a three-game series in Seattle beginning Monday.
NOTES: Griffin threw a second straight bullpen session without pain and could come off the disabled list Monday. ... RHP Ubaldo Jimenez (9-12) will take a second try at earning his 10th victory for Cleveland in Monday's series opener against Seattle. Jimenez has won at least 10 games in each of the previous four years. ... Oakland RHP Brandon McCarthy, coming off his first loss since April, pitches for the A's in Minnesota. McCarthy (6-4) is unbeaten in four career starts against the Twins.
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