Several names have been linked with the Selhurst Park hotseat since Ian Holloway departed last month, but Palace have vowed to take their time in order to ensure they get the right man for the job.
Dinamo Moscow coach Dan Petrescu has emerged as the latest favourite to take charge of the London club.
Parish admitted that the recruitment process has taken longer than the Premier League strugglers had hoped, but if all goes to plan an appointment will be made imminently.
He told Sky Sports News: "We're getting closer, 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.
"It's a difficult decision, important we get it right. 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 (on November 23).
"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 secret... one or two people abroad. All will be revealed soon."
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