Admin menu as submenu from another plugin

Trying to simulate the issue, it happened the same (wp-admin/submenu_slug), and the solution is to add a priority value in the hook admin_menu.

Here, I’m adding a sub menu to the plugin BackWPup. Note the priority 11:

add_action('admin_menu', 'third_party_submenu_wpse_91377', 11 );

function third_party_submenu_wpse_91377() 
{
    add_submenu_page(
        'backwpup', // Third party plugin Slug 
        'My plugin', 
        'My plugin', 
        'delete_plugins', 
        'third_party_submenu', 
        'plugin_options_wpse_wpse_91377'
    );
}

function plugin_options_wpse_wpse_91377() 
{ 
    echo '<h1>OK</h1>'; 
}