To grant image approval rights to non-supervisors, one of the roles they are a member of must have rights in a task group.  The most common one that Blackbaud NetCommunity site admins designate for this type of activity is "_Content Authors."

In order to grant the the permission, edit the SITE TASK group linked with the security assignment of the member role.  For instance, _Content Authors may have a task group called "_Content Authors (Site Tasks).

Add the "Approve photo rights" permission to the site task you are editing and click "Save". 

Have the user log out of Blackbaud NetCommunity and log back in again.  Permission to approve images should now exist.