Palace have been searching for a successor to Ian Holloway since last month, but Parish has insisted that they hope to make a decision as quickly as possible before their next Premier League fixture against Hull City.
Parish told Sky Sports News: "We are getting closer. It's quite fluid at the moment, but it has gone on a lot longer than we wanted it to.
"It's important that we get the right person. We have been working really hard on it, and talking to lots of people.
"Luckily, we have had the international break. We hope to have someone in by the weekend, so that they have a week to work with the players before the Hull game."