The Crystal Palace chairman allowed the manager to leave on Thursday, with the former Stoke City boss having grown frustrated at a lack of suitably talented targets
Pulis left the club by mutual consent on Thursday, just two days before their Premier League opener at Arsenal, which they went on to lose 2-1.
Goal understands that the Welshman's frustrations over signings were behind unsuccesful crisis talks on Thursday and, while Parish confirmed that the 56-year-old had been vexed by finding a lack of ideal targets to pursue, he is adamant that he did not push to sign players whom Pulis did not want.
"The stories about me wanting certain players and Tony not wanting certain players are just not the case," Parish told BBC Sport's '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 Pulis as a "wonderful coach", however, and said that the two parties owed each other nothing, now turning his attention to a successor, with an announcement hoped to be swift.
He added: "We want to get a manager who is right for the long term. 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."