this lnks to the update page are sending the ID using a url parameter named partner_id:
update_partners.php?partner_id=<?php echo $row_getPartner['partnership_id']; ?>
but the reopcrdset you have created on the update page, is execting a URL parameter named partnership_id.
either edit the links so they pass the URL parameter name the update page is expecting:
update_partners.php?partnership_id=<?php echo $row_getPartner['partnership_id']; ?>
or edit the recordset on the update page to expect the name you are using.