Crystal Palace sacked manager Patrick Vieira on Friday after a miserable run of one win in 14 Premier League games pulled them into a relegation battle.
The London club, without a win in 2023 under the former Arsenal and France midfielder, are 12th in the table but are just three points above the drop zone.
"Ultimately, results in recent months have placed us in a precarious league position and we felt a change is necessary to give us the best chance of retaining Premier League status."
Palace said in a statement that "the process to appoint a new manager is under way".
Three members of Vieira's coaching staff -- Osian Roberts, Kristian Wilson and Said Aigoun -- have also left Selhurst Park.
Parish added: "Patrick has given his all to the club, and we all thank him and his team for their service."
Vieira, 46, replaced Roy Hodgson before the start of the 2021/22 season and led Palace to a 12-placed finish.
Their 1-0 defeat by Brighton on Wednesday was their third loss in a row.
The Premier League's bottom nine clubs are separated by just five points heading into the final months of the season.
Palace travel to leaders Arsenal on Sunday.
Vieira, who won three Premier League titles as a player with the Gunners, previously managed New York City and Nice.
The Frenchman won the World Cup and European Championship in a glittering playing career.