The Eagles have been without a boss since Ian Holloway's departure on 23 October, with Aitor Karanka heavily linked with the post before joining Middlesbrough on Wednesday.
Palace have extended their search overseas for Holloway's successor, who will be tasked with lifting the club off the foot of the Premier League and out of the relegation zone.
"We're getting closer," Parish told Sky Sports News. "It's quite fluid at the moment. It's gone on a lot longer than we thought.
"We're talking to lots of people, getting lots of references. We are quite close."
Parish declined to give any names of those being considered, but said there were three applicants on the shortlist.
"It's a difficult decision, and it is important we get it right," he added.
"We thought we would have somebody by Tuesday or Wednesday. That hasn't transpired, but hopefully by the weekend, so they have a good week to work with the players before the (Hull City) game.
"Once we get to the point of making an offer, a decision is made. We are not talking to anyone where finances are a problem.
"It would be nice to do it on Saturday. It depends on complications - where they are physically, location. We've made no no secret... one or two people abroad. All will be revealed soon."