Chamakh has collected a hamstring injury.
Palace had been actively exploring the January transfer market for strikers and the news about Chamakh's injury intensifies their need to sign another forward.
"Chamakh is going to be out for a month so we will definitely look to strengthen there," said Parish to the London Evening Standard.
"We are always in a position to spend if we need to. We made a bid yesterday on one player from abroad and we are definitely looking at one or two positions. But we are confident in the group that we have got, so we are not desperate to bring people in."