hook for search a household contact with the surname individual field from online contribution page
If you're in Drupal, I would use Webform with Webform Civicrm Integration to collect more data. It allows you to collect info for multiple contacts as well as to create relationships between them. So for your use case, you could have for example a parent contact, a student contact, and a household contact, while all the relationships such as household member-household and parent-child get created automatically. 

If you're in wordpress, I'm not sure what the right plugin is, but I bet there is something similar you could use.