Parish also admits he would be interested in bringing Wilfried Zaha back to the club in the summer should the Eagles avoid relegation.
Pulis arrived at Selhurst Park on November 24 last year with the club second-bottom of the Premier League with just seven points from their first 12 games under Ian Holloway and Keith Millen.
His impact was immediate as they picked up back-to-back wins at Selhurst Park and Saturday's victory over Chelsea has left Palace five points above the drop zone ahead of Sunderland's Monday Night Football clash with West Ham.
When asked whether Pulis would be a contender for the award, Parish told Sky Sports News: "Why not? I don't know where we were when he came. We were probably bottom, but Tony would be the first to say that once we've achieved something then that would be a fantastic accolade.
"But he's very focused on trying to keep us up. After Saturday, the real key is to not get complacent. We know where we're at, we're not deluding ourselves because we've had one great result. We've got to keep on performing like that and we need a bit of luck."
Zaha was a high-profile departure from Selhurst Park last January, joining Manchester United in a £15m deal before being loaned back to Palace for the rest of the season.
But he failed to win a place in David Moyes' side this term and joined fellow strugglers Cardiff City on loan on deadline day at the end of January.
Parish said he would have no hesitation in bringing the England U21 winger back to the club if the chance arose in the summer.
"Wilf's a fantastic player, I certainly think we would be interested," he said. "United have not said anything official, he's their player, but if a player like that did become available then I am sure we would be interested.
"If we are fortunate enough to stay in the division, we will be looking to kick on and establish ourselves and try to get up to mid-table and be a bit more comfortable.
"Wilfried is a player who is worthy of the division and is a player who could help anybody. We saw him come on for Cardiff at the weekend and he changed the game, so if a situation arose for that kind of player, of course we would be interested."