Arizona trailed 1-0 when Parra hit a drive to right-center. The ball hit off the heel of McLouth's glove as he ran into Heyward.
McLouth spun around and fell backward, his head slamming on the ground. He stayed there face-down for several minutes before he was able to get up and walk off the field.
Heyward got up and tracked down the ball to start a relay to the plate that was too late. It was Parra's first career inside-the-park homer and the seventh in Diamondbacks history.
The Braves said McLouth had a headache and would be evaluated through the night.
Peter Moylan (2-1) hit Mark Reynolds with a pitch to lead off the Arizona eighth inning. The one-out homer was the first off Moylan in 123 appearances, a span of 102 1-3 innings.
Ian Kennedy blanked Atlanta on three hits through seven innings and Kenshin Kawakami gave up four hits in seven innings for the Braves.
The hard-luck Kawakami remained 0-8, the first Braves pitcher since Rosy Ryan did it for the Boston Braves in 1925. Chad Qualls pitched a scoreless ninth for his 12th save in 16 tries.
Doubles by Jason Heyward and Brian McCann off Aaron Heilman (2-1) had put Atlanta up 1-0 in the eighth.
In Arizona's eighth, Reynolds was hit and advanced to second on Chris Young's sacrifice bunt.
Parra, who had not homered this season, lofted a high fly. McLouth, who had made a brilliant, diving catch earlier in the game, never took his eyes off the ball as he ran it down. Neither, apparently, did the 6-foot-4, 220-pound Heyward, who undercut the Atlanta center fielder as the ball bounced away toward the fence. By the time Heyward chased it down, Parra was rounding third and headed home.
Both teams wasted opportunities.
The Braves had a chance to give Kawakami the lead in the seventh when Omar Infante's pinch-hit single put runners at first and third but Kennedy struck out Martin Prado to end the inning.
Atlanta loaded the bases with one out on two walks and a single in the second but McLouth struck out and Kawakami grounded out.
Reynolds and teammate Justin Upton kept up their heated battle for the major league strikeout lead. Upton fanned three times and Reynolds twice, leaving the two tied at 80.
Atlanta's Troy Glaus was 0 for 3, all strikeouts.
NOTES: Atlanta RHP Kris Medlen said his left shoulder was sore but otherwise fine a day after he strained it diving headfirst into home plate. He expects to make his next schedule start. ... LHP Dontrelle Willis makes his second start for Arizona on Thursday after throwing six sometimes-shaky but scoreless innings Saturday in his Diamondbacks' debut, a 4-3 win over Colorado. Tommy Hanson (6-3) goes for the Braves. ... Atlanta has left 34 runners on base in the last three games, eight on Wednesday.