Add parent menu item in sub-menu in custom nav_walker

So you need to real name and URL of the parent in the submenu.

You have no info about a post item in your function start_lvl().

But in function start_el() you have the items and could query for it’s parent.

This needs some refactoring with putting lines like the if($display_depth > 0){ condition from start_lvl function into your start_el function. Then writing a proper WP query to get the Info’s of the parent item you need in the submenu.

To point a bit more specific:

In start_el function you have the var $item which should contain the ID of the post it represents. Should be $itemId = $item->Id.
If so you should get the parent with $parentId = wp_get_post_parent_id( $itemId ); and further more you can fetch infos from the parent with the parent ID with some other WP functions.

Hope that helps to find your way..