Tyler Skaggs (1-0) pitched 6 1/3 innings of three-hit ball in his first win over the A's in six career starts. The left-hander, who missed 85 games last season due to a strained oblique, allowed only one runner past first base.
It was Los Angeles manager Mike Scioscia's 1,571st win, tying him with Hall of Famer and former A's and Angels manager Dick Williams for 21st all-time.
Parker nearly let it slip away in the ninth.
He issued a leadoff walk to cleanup hitter Khris Davis and Matt Olson followed with a single to right. After striking out Stephen Piscotty and Matt Chapman, Parker allowed Jonathan Lucroy's RBI single before he got Matt Joyce to ground out to second for the save.
Japanese two-way player Shohei Ohtani, who singled in his first at-bat in the majors on Thursday, was held out because the A's started a left-hander. Ohtani is scheduled to make his pitching debut Sunday.
Trout homered off Sean Manaea (0-1) in the first. He also reached on a fielder's choice in the ninth and scored from first when left fielder Joyce short-hopped Justin Upton's single and the ball rolled to the fence.
It was Trout's 26th career home run against the A's and 14th at Oakland Coliseum.
Skaggs and the bullpen made it stand up after Los Angeles lost on opening day in 11 innings after blowing a four-run lead.
The Angels also helped themselves with a pair of sparkling defensive plays.
Manaea allowed four hits over 7 2/3 innings with seven strikeouts. He retired 12 straight and 20 of 21 after Trout's home run.
The A's stranded three runners at third.
Angels: Leadoff hitter Ian Kinsler started at second base after sitting out the final two games of spring training and Thursday's opener with a groin injury. . Andrew Heaney, on the disabled list with left elbow inflammation, threw a bullpen session before the game.
Angels: RHP Matt Shoemaker pitches Saturday in his first regular-season start since going down with a season-ending forearm strain June 14.
Athletics: RHP Daniel Mengden is coming off an injury-plagued season after being limited to seven starts a year ago.