The outcome denied the Twins (27-27), who batted back from a 3-0 deficit, a share of first place in the West Division.
Mason Berne opened the home half of the ninth with a single, Juan Morales laid down a sacrifice bunt, then Garrett Saunders was hit by a pitch. A Jose Palma groundout advanced Berne to third and Saunders to second, setting up Parker for a two-out hit.
Will Holland got Elizabethton on the board with a sixth-inning home run. The Twins caught Danville (21-34) in the eighth when Kidany Silva lined an run-scoring single and Anthony Prato delivered a sacrifice fly.
Both teams pounded out 11 hits. The Braves got two apiece from Palma (2 runs), Parker (2 RBIs) and Morales.
Ruben Santana had a strong game from the top of Elizabethton’s order, going 4 for 6. Max Smith (3 for 4, walk) and Prato (2 for 3, double) also excelled from an offensive standpoint.
With 1 1/3 innings of scoreless relief, Justin Yeager (2-1) closed out for the win. Lisandro Santos blanked the Twins over the first five innings.
Elizabethton reliever Ryan Shreve hurled four shutout innings before Osiris German (3-1) came along in the ninth to suffer his first setback.
The two teams meet again on Friday, starting at 6:30 p.m.