The 34-year-old goalkeeper, who joined the Eagles 10 years ago, is due to be a free agent this summer.
However, Parish has suggested that the club is eager to keep the Argentine at Selhurst Park beyond the end of his playing career.
"There is no date set but I think everyone wants a little break," he told the Croydon Advertiser.
"Tony [Pulis] is going to have a chat with Julian and talk to him about where he sees him and how he sees it going forward.
"We would all like to keep Julian at the club, of course we would. It's looking for a long-term plan as well. I think we would like to see Julian really staying at the club and having a future, maybe on the coaching side."
Speroni recently revealed that he is keen to sign a new deal with Palace.