If you want to build your own custom nav menu you should be extending the Walker_Nav_Menu class, not the Walker class.
When building a custom menu walker the $db_fields should be what the default Walker_Nav_Menu class has:
$db_fields = array ('parent' => 'menu_item_parent', 'id' => 'db_id');
When building the navigation link for a menu item, the parent field maps to the _menu_item_menu_item_parent post meta field if the menu item has a parent.
The db_id maps to the post ID of the menu item.
Check out wp_setup_nav_menu_item() for good reference to better understand what is happening. https://developer.wordpress.org/reference/functions/wp_setup_nav_menu_item/