Tajouri-Shradi's score off a corner kick, his second of the season, came four minutes after midfielder Nicolas Acevedo was sent off following a reckless tackle. The goal came after Alfredo Morales headed a pass to the far post where Tajouri-Shradi scored with his left foot.
After a tight first half with few legitimate scoring chances, LAFC broke through first in the second half. Mark-Anthony Kaye delivered a centering pass from the right end line that was deflected by NYCFC defender Alexander Callens.
The ball rebounded to the front of the NYCFC goal where Baird was able to tap it in for his third of the season.
NYCFC needed 14 minutes to get even after four players entered the game, including Tajouri-Shradi, Anton Tinnerholm and 18-year-old Talles Magno, who was making his debut with the club.
Tinnerholm picked up a loose ball in the LAFC penalty area and fired a quick centering pass to Medina, who delivered the equalizer. It was Medina's fifth goal of the season, matching his total in 22 games (18 starts) last season.
Magno, who was acquired last week on an $8 million transfer from Vasco da Gama in Brazil, played 24 minutes with his time limited as he recovers from a knee injury.