Captain Parejo set up Alcacer for the opener and then scored a superb free-kick on the stroke of half-time after Augusto Fernandez had equalised for Celta, who had lost only one previous league game this season.
Jonny Castro's poor backpass gifted Alcacer another early in the second half and he then laid on the fourth for Parejo, who in turn became the provider for Shkodran Mustafi to round off an emphatic victory.
Levante came from behind to draw 1-1 with Deportivo La Coruna and earn their first point under new coach Rubi.
Lucas Perez had given the visitors the lead midway through the first half against the run of play but Levante's energetic approached paid off after half-time, with Victor Camarasa levelling with a free-kick.
Sergi Enrich scored twice as Eibar continued their march up the table, a 3-1 win over Getafe taking them sixth.
Enrich's double sandwiched Victor Rodriguez's reply, and Saul Berjon made the points safe with a penalty after Damian Suarez had been sent off.
Javi Guerra took his recent goal tally to eight in as many games with a double as Rayo Vallecano beat Granada 2-1 despite the 19th-minute dismissal of Ze Castro. Jean-Sylvester Babin scored Granada's goal.
In the day's late game, Real Betis won 1-0 at Malaga to jump to ninth place. Ruben Castro scored the only goal in the 64th minute.
