Parejo gave Valencia the lead from the spot on 19 minutes and doubled their advantage on the half-hour mark when he slid a low shot past Sevilla goalkeeper Beto.
Parejo then gave Sevilla a route back into the match in the 36th minute when he tripped Alvaro Negredo in the area and Carlos Bacca made no mistake from 12 yards.
Sevilla were awarded another penalty five minutes later but Bacca was denied as Valencia stopper Diego Alves saved with his legs.
Javi Vuego made sure of Valencia’s victory shortly before the hour mark when he unleashed a ferocious shot into the bottom corner from the edge of the penalty area.
Victory saw Valencia move ahead of Unai Emery’s side and into fourth spot in La Liga.
New Granada coach Abel Resino saw his side snatch a late point away to relegation rivals Deportivo La Coruna as his first match in charge ended in a 2-2 draw.
Piti had given Granada a sixth-minute lead but Depor turned the game around with two goals in the space of five minutes from Jose Rodriguez and Lucas Perez, and they seemed well set for the three points.
However, Robert Ibanez struck with seven minutes left and Depor finished the match with 10 men after Luis Farina was sent off in the closing stages.
Javi Guerra’s strike 12 minutes from time ensured Malaga claimed a 1-1 draw at Atletico Bilbao, whose four-game losing streak in the Primera Division was brought to a halt.
Guerra’s strike came just eight minutes after defender Mikel San Jose had given Bilbao the lead.
Espanyol rounded off a fine week with a 3-0 win over Almeria, with Cristian Stuani grabbing a first-half double before Felipe Caicedo wrapped up the points.