The Eagles were second bottom of the table and had only won two of their opening 12 games when Pulis took the helm following Ian Holloway's decision to resign.
Parish has admitted that their escape mission is far from finished, but has suggested that, should he successfully steer them away from the drop, Pulis's achievements would be worthy of recognition.
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."
Palace have lost only three of their 10 home games since Pulis was appointed in December.