Give a User Rights to Edit Parse Table Row (Class)

Basically if you save the user with the record then they have write access to that record.

JAVA

ParseObject myParseTable = new ParseObject("MyParseTable");
myParseTable.put("ID", "1");
myParseTable.put("Message", "I can now edit this message.");
myParseTable.put("author", ParseUser.getCurrentUser());

myParseTable.saveInBackground();

SWIFT

var myParseTable = PFObject(className:"MyParseTable")
myParseTable["Message"] = "1"
myParseTable["Message"] = "I can now edit this message."
myParseTable.ACL = PFACL.ACLWithUser(PFUser.currentUser())

myParseTable.saveInBackground()