Odd behaviour with submenu link creation

On the same page in codex you have this :

NOTE: If you’re running into the “You do not have sufficient permissions to access this page.” message in a wp_die() screen, then you’ve hooked too early.

So it answers to you question partly.

The second part, the following code should work :

add_action('admin_menu', 'my_add_submenu');
function my_add_submenu(){
      add_menu_page( 'Parent', 'Parent', 'manage_options', 'author_discuss', 'author_parent' ); 
      add_submenu_page( 'author_discuss', 'Author Discussion', 'Author Discussion', 'manage_options', 'author_discuss' );
      add_submenu_page( 'author_discuss', 'Author Discussion Settings', 'Settings', 'manage_options', 'author_discuss_settings', 'author_discuss_settings_page' );
}

function author_parent() {

}

function author_discuss_settings_page() { 
    //do something
}

Check you functions and see if there’s some typo.