The two sides went head to head at Anfield in a match that Jurgen Klopp's side dominated from start to finish in terms of possession and chances created. However, they could only manage one goal through Egyptian ace Mo Salah.
It gave Everton hope and Allardyce's side earned a 1-1 draw following a clumsy challenge by Dejan Lovren on Dom Calvert-Lewin in the penalty area, which saw Wayne Rooney dispatch a spot-kick past Simon Mignolet with ease.
Aldridge, who played for Liverpool for two years between 1987 and 1989, claimed that Allardyce is ego-driven and stated that Everton fans will soon get bored of his style of football.
"Everton were clearly unsure about appointing Allardyce when they initially held talks with him and the club’s supporters pretty much united in their stance that they didn’t want him.
"This guy promotes a brand of football that is hard to watch and while it gets results and keeps average teams in the Premier League, fans get bored of it pretty quickly and that will happen again at Everton."
Everton remain unbeaten under Allardyce since the former England boss took over at Goodison Park, earning wins against Huddersfield and Apollon Limassol as well as drawing to the Toffees' bitter rivals.
However, sitting way down in tenth place, they remain a huge 11 points behind Liverpool despite pre-season expectations suggesting they could rival the Reds for a place in the top four.
Liverpool and Everton go head to head in the FA Cup third round on January 5, before meeting for a third time on April 7 in the return Premier League fixture.