That was Parker's last batter, but Crawford would come around to score on Brandon Belt's ensuing double against Minor Leaguer Ryan Dull.
Lefty Sean Doolittle had an easier time with San Francisco's lineup, striking out two in a perfect third inning in his spring debut.
The Giants, meanwhile, got two clean innings from lefty starter Edwin Escobar. San Francisco's pitching prospect fanned one in his debut.
The A's didn't get on the board until the fourth inning, when Sam Fuld led off with a triple and scored on Jed Lowrie's sacrifice fly. In the fifth, Stephen Vogt collected his second hit, an RBI single that plated Daric Barton, who led off with a double.
Jake Elmore, acquired by the A's on Thursday, made his debut in the green and gold, and singled in his first at-bat and scored a run.
Oakland's Derek Norris hit a two-run homer in the eighth to tie the game at 6, before Heath Hembree's wild pitch allowed the winning run to score in the bottom of the ninth.
Up next: Intent on sustaining the momentum he built in last season's second half, Matt Cain will make his first Cactus League start when he faces the Chicago Cubs on Saturday in Mesa, live on Gameday Audio at 12:05 p.m. PT. Cain fashioned a 2.36 ERA in 11 starts after the All-Star break, compared with 5.06 ERA in 19 outings beforehand.
Jane Lee is a reporter for MLB.com. Read her blog, Major Lee-ague, and follow her on Twitter @JaneMLB. This story was not subject to the approval of Major League Baseball or its clubs.