Pulis quit the club suddenly on Thursday, just days out from their Premier League opener at Arsenal.
Reports suggested Pulis was unhappy with Palace's failure to sign transfer targets, but Parish said that was untrue.
"The stories about me wanting certain players and Tony not wanting certain players are just not the case," Parish told BBC Match of the Day on Saturday.
"My job is to try to get the deals done for Tony or any other manager.
"There was a good transfer budget agreed. There was a lack of targets the manager felt were good enough in the window and it came to a point where he felt he could not carry on.
"I can't have people at the football club who are not 100 per cent committed to what we are trying to do, so reluctantly I agreed to let him go and let him out of his contract.
"It is very important we have a good communication and for me the communication was difficult."
Parish praised "wonderful coach" Pulis and said the two parties owed each other nothing.
Speaking after Palace's 2-1 loss to Arsenal, Parish said he wanted to have a replacement announced no later than Wednesday.
"We want to get a manager who is right for the long term," he said.
"Experience of the Premier League is very important and we would not be straying too far from someone who has that intimate knowledge of it, either as a player or as a manager.
"Hopefully Tuesday or Wednesday this week we will try and get someone in place."
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