The Eagles have been slow out of the blocks in the transfer window, only bringing in Fraizer Campbell, free-agent defender Brede Hangeland and young Scottish goalkeeper Chris Kettings so far this summer.
However, Parish has suggested that further arrivals could be imminent, including a mystery midfielder who is expected to undergo a medical and discuss personal terms ahead of switching top-flight clubs.
When asked about the mystery midfielder, Parish told Sky Sports News: "We have a bid in for a player tonight. We've had permission to talk to the player. We are very close to personals with the player. Hopefully he will be down for a medical tomorrow and with any luck we'll get him in the squad for Saturday.
"It's another really good moment for our football club and a good moment of advancement. A very good player, someone we are looking forward to welcome at the football club. We've not signed on the dotted line yet, I don't want to say who it is, but we're very excited to get it done."
Palace travel to the Emirates Stadium to face Arsenal in their opening Premier League fixture on Saturday.