Parejo saw his late penalty saved by Ivan Villar on Monday night but converted the rebound to secure the win in a game where the home side twice came from behind and then gave up the lead before snatching the victory at the very end.
The result lifts Villarreal to the top of La Liga with seven points from three games, one ahead of Celta and Barcelona, who have played one match less.
Celta were one of only two teams to win their first two league games and went ahead after 11 minutes when Borja Iglesias took advantage of sloppy goalkeeping, but Sergi Cardona equalised 15 minutes later by slamming home a corner that was nodded across goal by Gerard Moreno.
Oscar Mingueza continued his recent fine form by putting Celta back ahead after half an hour with a low 30-yard free kick.
Then new signing Thierno Barry, who joined last week from FC Basel in Switzerland, headed home with his first touch of the ball only a minute after coming on as a substitute.
An own goal from Jailson gave Villarreal the lead until Carl Starfelt made it 3-3 with 10 minutes remaining.
Both sides pushed for a winner and Celta hit the post before the referee awarded a penalty to Villarreal for a foul on Barry.
"If you have to lose then that's the way," Iglesias said.
"The fans must have enjoyed that game."
