Three of the Reds' runs came at the expense of Arizona's Tyler Skaggs, who is battling Patrick Corbin and Randall Delgado for the No. 5 spot in the rotation.
While Skaggs did not help himself by allowing three runs on three hits in just two-thirds of an inning, Reds infielder Jason Donald did.
Donald went 3-for-5 with two doubles and a pair of RBIs as he fights for a utility spot on the Cincinnati roster.
Ryan Ludwick and Donald drove each drove home a run in the third, which also featured a balk and wild pitch by Skaggs.
The D-backs grabbed a 1-0 lead in the first when Willie Bloomquist doubled to left and scored when Parra followed with a grounder back up the middle.
The Reds capitalized on a two-base error by shortstop John McDonald to tie the game in the second.
With one out, Todd Frazier reached second on the error and Donald followed with a double to left.
Reds starter Mike Leake allowed one run on two hits in two innings of work, with three strikeouts, as he tries to outduel Aroldis Chapman for a spot in the Cincinnati rotation. Chapman tossed two scoreless innings in the Reds' other game Thursday, against Colorado.
Arizona Opening Day starter Ian Kennedy tossed two innings, allowing two hits and one unearned run, with one strikeout. Paul Goldschmidt and Mark Teahen each had an RBI for the D-backs.
Up next: Mat Latos will be making his first Cactus League start when the Reds travel to Surprise and face the Royals at 3:05 p.m. ET on Friday. Several key relievers are due to follow Latos, including Sean Marshall, Jonathan Broxton and Sam LeCure.
Steve Gilbert is a reporter for MLB.com. Follow him on Twitter @SteveGilbertMLB. This story was not subject to the approval of Major League Baseball or its clubs.