The Premier League's bottom-placed side have been without a boss since Ian Holloway's departure last month, with Keith Millen taking charge on a caretaker basis since then.
A host of coaches have been linked with the Selhurst Park hotseat, including Tony Pulis and Dan Petrescu, and Parish has confirmed that an appointment could be imminent.
"Hopefully we'll get somebody announced tomorrow (Monday), but we can't be sure," he told talkSPORT.
"It's still a little bit up in the air, but the most important thing is we're doing our best to get the right person."
Parish went on to dismiss talk that Palace - who have taken just four points from their 11 Premier League games this term - have taken too long to decide on Holloway's permanent successor.
"Selecting a new manager is a pivotal decision, it's the most difficult one we've ever made," he added.
"We want to find someone who can help us stay up and we need someone for the long term, if the worst happens and we go down.
"We've seen and considered just about everybody, which I think is important."
Should Parish's timetable prove correct, the new Palace boss' first assignment will be a trip to Hull City on Saturday.
