Ryu, 32, was five outs away from history when Parra doubled. Parra signed with the Nationals earlier this week and had a go-ahead grand slam Saturday night.
Parra moved to second on Wilmer Difo's sac bunt before Ryu induced a lineout from Michael A. Taylor.
Ryu struck out nine and walked only one in the Dodgers' 6-0 win. He is now 5-1 with a 1.72 ERA in eight starts this season.
Oakland's Mike Fiers pitched a no-hitter Tuesday against the Cincinnati Reds.
Four Dodgers pitchers combined for a no-hitter against the Padres on May 4, 2018. Clayton Kershaw threw the Dodgers' last traditional no-hitter in June 2014.
Los Angeles moved to 27-16 with Sunday's win. The Dodgers lead the NL West Division by 3.5 games.