Parejo opened the scoring from the penalty spot after Diogo Figueiras was adjudged to have fouled former Sevilla striker Alvaro Negredo and the midfielder doubled the home side's lead 32 minutes in.
Carlos Bacca gave Sevilla hope with a spot-kick four minutes later, but penalty-saving specialist Diego Alves produced yet another stop from 12 yards to deny Bacca an equaliser from 12 yards before half-time.
There was no way back for the Andalucian outfit and Javi Fuego struck Valencia's third goal as they extended their unbeaten run to seven Liga games and moved two points above Sevilla.
Abel Resino was unable to mastermind a victory in his first game back in charge of Granada, but Robert Ibanez's late strike at least salvaged a 2-2 draw at Deportivo La Coruna.
Resino returned for a second spell as coach of the bottom side in La Liga this week and Piti raised hopes that he could start his reign with a win when he opened the scoring after seven minutes.
Granada had not won in 15 games prior to Resino's arrival, though, and two goals in the space of four first-half minutes from Jose Rodríguez and Lucas Perez saw them fall behind.
However, Depor were unable to hang on as Ibanez rescued a point before the home side had Luis Farina sent off in stoppage time.
Christian Stuani scored twice and Felipe Caicedo was also on target as Espanyol consigned struggling Almeria to a fourth consecutive defeat in all competitions, winning 3-0 on home soil.
Athletic Bilbao's winless run was extended to five matches courtesy of a 1-1 home draw with Malaga - who are also without a win in five - in the other game to be played on Sunday.
