Parker (2-2), making just his ninth major-league start, did not allow a base hit until Michael Young led off the eighth with a single to center field.
The right-handed former first-round draft pick finished eight full innings having allowed just the one hit, while issuing three walks and striking out six for the A's, who took the opener of a four-game series against the American League West leaders.
Brandon Inge triggered an eight-run second inning with a three-run homer and added a fourth RBI with a run-scoring single later in the frame. The A's sent 13 men to the plate during the big inning.
Kila Ka'aihue socked a two-run homer and Seth Smith went 4-for-5 with an RBI for Oakland.
Rangers starter Scott Feldman (0-4) was rocked for eight runs on seven hits, lasting only 1 2/3 innings.