Palace got rid of former Newcastle boss Pardew after slipping down the table.
The Eagles are just one point and one place above Sunderland, who occupy the final relegation spot, after Pardew guided the club to just four wins in the first 17 games of the season.
And ex-Black cats boss Allardyce – who left Sunderland in the summer after guiding them to safety – has emerged as the early frontrunner for the job, with Paddy Power installing him as 4/7 favourite.
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Allardyce stepped down as England manager in September after newspaper revelations.
In a statement on their website, Palace chairman Steve Parish said: “I would like to put on record our sincere gratitude for the tremendous service Alan has given us, both in his time here as manager and previously as a player.
“Following a fantastic second half of 2014/15, the 2015/16 season culminated in only narrowly missing out on winning the FA Cup as well as securing our Premier League place for a record breaking fourth time.
“During his tenure Alan’s hard work and dedication were without question, he has improved so many areas of the club and we wish him the very best for the future.
“With games now coming thick and fast over the holiday period the Club are hoping to put someone in place as quickly as possible but have nothing to say about a replacement at this time.”