If you have existing links that refer to the page using the old parent’s slug, they’ll need to be updated to reflect the new parent. The nav menu links should update automatically if the menu items are actual page items. Custom items need to be manually updated. You can use the Better Search and Replace plugin to seek out and correct all links with the old parent slug.
If there are external links with the old parent slug, they may not be so easy to update. You may want to add a specific redirect directive in your .htaccess file to take old parent slug requests and 301 redirect them to the new URL.
