It is now three weeks since Holloway left Selhurst Park by mutual consent following the Eagles' tough start to life in the Barclays Premier League.
Palace have secured just one point from their three games since, although Parish is confident of having a new man in place for the trip to Hull after the international break.
"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."
Another reported target was Wales manager Chris Coleman, although Press Association Sport understands the former Palace defender has signed a contract extension to remain in charge for the Euro 2016 qualifying campaign.
Former Inter boss Andrea Stramaccioni and, most surprisingly, ex-Palace boss Iain Dowie remain reported possibilities for the south Londoners, although Dynamo Moscow manager Dan Petrescu is the bookmakers' favourite.
The 45-year-old spent time at Sheffield Wednesday, Chelsea, Bradford and Southampton during his playing career, before turning his hand to management.
Dynamo is Petrescu's seventh club of a decade-long coaching career and Parish was coy when asked about those being considered.
"It's a difficult decision, important we get it right," he said. "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] 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 secret... one or two people abroad. All will be revealed soon."
While a new manager is not yet in place, Palace did manage to bring in former Cardiff head of recruitment Iain Moody at the weekend.
Moody left his position at the Cardiff City Stadium last month and was replaced by 23-year-old Kazakh Alisher Apsalyamov, in a surprise move by owner Vincent Tan.
However, he has quickly found himself back in the Premier League, joining Crystal Palace as their new sporting director.
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