First of all, I realize this is nitpicky, but it kind of matters, I highly suggest you follow Java naming conventions. Your classes should start with a capital letter (i.e. GameChat and GameLobby).
'This' does not refer to the superclass. 'This' refers to the current instance of that object, so in your case the current instance of 'gameChat'.
Calling getParent() has nothing to do with the object oriented structure of your program. Don't confuse Swing's layout system with your Object hierarchy!
getParent() returns the parent container of the component. It's probably null for a JApplet.
You get the Applet from another class simply by passing it in a method or constructor. You can then call any of its methods, including setEnabled(boolean).
Example 1
public class SomethingElse {
public SomethingElse(gameLobby lobby) {
lobby.setEnabled(false);
}
}
Example 2
private static gameLobby inst;
public gameLobby() {
inst = this;
}
public static gameLobby getInstance() {
return inst;
}