The Senegalese defender was involved in a car accident on the M4 near Heathrow on Sunday and has undergone surgery on his thigh and jaw.
Parish says the 26-year-old left-back is expected to make a full recovery.
"Thanks to the fantastic response of the emergency services who cut him from the car and the amazing surgeons, doctors and staff at the Royal London Hospital," Parish said in a club statement on Palace's official website.
"He is expected to make a full recovery and should be discharged on Friday.
"We would like to make special mention of the London Air Ambulance service who flew Pape directly from the scene to the hospital ensuring that he rapidly had access to the very best medical care.
"The club note that this organisation is a charity and completely funded from donations as such we are making a contribution and would urge everyone to support this invaluable service for the people of London."