The Welshman has been without a club since leaving Stoke at the end of last season and has emerged as the favourite for the vacancy at Selhurst Park after Ian Holloway's exit
Pulis has been out of work since leaving the Britannia Stadium at the end of last season and was linked to Sunderland after the sacking of Paolo Di Canio, before the Black Cats announced Gus Poyet as manager.
Parish has revealed that Pulis, who kept Stoke in the Premier League for five consecutive season before parting with the Potters, has the experience Palace are looking for in their new manager.
"There are some obvious names and to not have a conversation with them, if they’re happy to do that, would be silly," Parish told TalkSPORT. "So, I’m sure we’ll try and contact him [Pulis] and if he’s interested we’ll have a chat with him and see where we go.
"I think, Tony has never got relegated has he? He’s out of work, got Premier League experience. Not a bad shout.
"But I don’t want to put pressure on anybody. We’ve started making a list, we’ll throw everybody into that and see who we want to see."
Holloway's decision to leave came after he decided that someone else could do a better job of helping the Eagles avoid relegation, with the south London club second-bottom of the league.
"There are catalysts that happen. Sometimes success and failure are the same imposter," Parish added.
"I know Ian’s going because he feels we could do better with someone else, if he’s right then we could look back and go 'cor you know we didn’t want him to go and yet he went and equally we could look back and feel we should have tried and almost made him stay."