Luis Barraza registered five saves for NYCFC (3-2-3, 12 points), who ended a three-match winless stretch (0-1-2).
Reigning MLS MVP Hany Mukhtar scored the goal for Nashville SC (3-3-2, 11 points).
Nashville allowed two goals in the first 25 minutes in Saturday's match after giving up just two total over its first seven matches.
Nashville goalkeeper Joe Willis, who has five shutouts this season, made four saves.
Nashville made a late charge when Muhktar found the back of the net in the 81st minute. Daniel Lovitz had the ball on the left wing and passed it back toward the center of the box and Mukhtar ripped a right-footed shot into the lower left corner.
Mukhtar had a right-footed shot blocked one minute later and Shaq Moore's right-footer in the 84th minute was wide to the right.
The contest completed the two-game regular-season series. Nashville notched a 2-0 home win in the season opener on Feb. 25. NYCFC improved to 1-2-1 all-time against Nashville.
Nashville held a 15-14 edge in shots and both teams placed six on target.
NYCFC struck in the 11th minute as Mitja Ilenic sent in a cross and Parks knocked a header past Willis for his first goal of the season.
NYCFC made it 2-0 in the 25th minute as Richard Ledezma took a corner kick and sent the ball directly to Chanot, who delivered a right-footed volley shot past Willis. The goal was Chanot's first of the campaign.
Nashville had a chance to get on the board in the 42nd minute but Barraza made a diving hand save on a right-footed shot by Mukhtar.
In the 63rd minute, Nashville's C.J. Sapong took a left-footed shot that Barraza saved.
Moore kicked in a rebound in the 77th minute but was ruled to be offside and the goal was waved off.