Palace have been searching for a boss since the shock departure of Pulis on the eve of the season and only Malky Mackay and Tim Sherwood have been interviewed for the job, with Keith Millen currently in caretaker charge.
Mackay is no longer being considered after he was caught up in allegations of racism and homophobia brought by former club Cardiff, while Sherwood pulled out of the running last week.
Neil Lennon, Ally McCoist and Steve Clarke are the latest names to be linked to the vacant role and Parish remains hopeful he will find a successor soon, but is yet to hold any further interviews.
He told the Croydon Advertiser: “You read all sorts of things about how many people we’ve seen.
“I have seen very few people, probably two, but spoken to other people about managers and doing background work and due diligence.
“It’s not a revolving door of people coming in and having a chat. We have got a profile of a manager we think needs to carry the club forward based on what I’ve experienced over the last four years.
“Finding the right person isn’t easy. A lot of managers are in jobs and they’re not going to leave clubs early in the season, so it’s a difficult situation.”
Palace have made only a handful of summer signings and injuries and suspension took its toll on the team which slumped to a 3-1 defeat at home to West Ham on Saturday.
The new man at the helm is expected to be handed a transfer budget of around £25m, but Parish accepts he must act fast before the deadline window closes next Monday.
“We have got to get the right answer and it’s an ongoing process,” he said. “Obviously I hope to try and have someone in the early part of this (next) week, because we have got the transfer window.
“I may have to start picking the players who come to the club! Keith and I were joking about that the other day - I need a manager because they will know which players he wants.
“We buy the players that the manager wants at this football club. We research them but the manager has the decision on who comes.
“We have got a week of the transfer window left so it’s important we get a decision to see who is going to run the place.”