Pulis stunned Palace by asking to leave his role as manager just two days before the season started, with Parish agreeing to his request.
Former Stoke boss Pulis - who guided Palace to a miraculous escape last season - was reportedly unhappy with the way the club had done their transfer business in the summer.
But although Parish is sad to see the Welshman move on, he is adamant he did the right thing by not stopping him from walking.
"I can't explain it, I'd love to be able to tell you everything that went wrong," he told Match of the Day. "There was a good transfer budget agreed, there was a lack of targets that I think the manager felt was good enough in the window and it came to a point where he felt he didn't want to carry on.
"I can't have people at the football club who aren't 100 per cent committed to what we're trying to do, so reluctantly I agreed to let him out of his contract.
"I can't be let down for what he achieved for us. He doesn't owe us but we don't owe him. He did a fantastic job for us but we are what we are. I got this club from administration to where we are now by taking decisions to create a positive environment.
"If people don't feel they can work in that environment then it's fine for them to move on."
Parish also thinks that his old boss will be back working again soon.
"You'll have to ask him but I think he intends to work. You have to assume that if he doesn't want to be at this club there's one that can give him the things that he wants and can make him happy."
Meanwhile, Palace caretaker boss Keith Millen said he wanted the job on a full-time basis after his resilient team made Arsenal scrap all the way before snatching a 2-1 victory on Saturday.
"I'd like to continue, I enjoy being in charge," Millen said after Palace narrowly lost their opening game of the season.
"Whether it's the right time or not I'll speak to the chairman tomorrow.
"If the chairman felt it was right for me then I would sit down and talk to him about it. I like working at this club."
Millen also served as caretaker for a short spell last season following Ian Holloway's exit. He distanced himself from the full-time role on that occasion, citing inexperience.
"The circumstances of last year were that we had a really poor start, winning one in 10 games," Millen explained.
"There was a lack of Premier League experience at the club. At the time I felt someone like Tony could give us that. This group of players now have that experience."
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