Aprove answer
var user = PFUser.currentUser()
user.setValue(newEmail, forKey: "Email")
user.saveInBackgroundWithBlock {
  (succeeded: Bool!, error: NSError!) -> Void in
       if error == nil {
           println "Profile Updated."
       } else {
           println "Failed"
           //present alert to user to let them know that it failed
           //ask them to try a new email address

Credit Goes to: stackoverflow.com

Related question with same questions but different answers