Celta opened the third round of the Liga season top of the table on goal difference from Real Madrid after winning their opening two games.
Borja Iglesias, making his Celta debut, gave the visitors the lead when he pounced on a loose ball and tucked it into an empty net after 12 minutes.
Sergi Cardona, unmarked at the far post, levelled from a 26th-minute corner.
Oscar Mingueza smashed Celta back ahead before half-time.
Villarreal substitute Thierno Barry, who had been on the field for a minute, headed an equaliser on the hour.
Four minutes later, Celta defender Jailson turned a cross from Nicolas Pepe into his own net to put Villarreal ahead.
Swedish centre-back Carl Starfelt headed Celta level in the 80th minute.
Deep into stoppage time Celta's Hugo Alvarez, a second-half substitute, pulled down Barry to concede a penalty.
Goalkeeper Ivan Villar saved Parejo's spot kick but the midfielder scored from the rebound.
The victory took Villarreal to seven points and they took over top spot.
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