The hosts were playing for the first time since lifting their first major trophy in 40 years with victory last weekend over Real Mallorca in the Copa del Rey final on penalties.
It appeared as though fortune from the spot would be on their side again as Villarreal striker Gerard Moreno struck the post with an early penalty.
Visiting midfielder Santi Comesana was dismissed for a second yellow card in the first minute of the second half and Athletic took advantage when Oihan Sancet fired home in the 67th minute.
But a VAR review resulted in another Villarreal spot-kick in the fifth minute of added time, and Parejo made no mistake.
The Basque club now sit four points behind fourth-placed Atletico Madrid in the race to reach the Champions League with seven matches remaining.
