My parents received an email from Orcon yesterday afternoon that states their email account will be terminated in one month because it has not been accessed in over 12 months. 

This is absolutely incorrect - the account is used all the time.

I called the helpdesk and was told to ignore the email. I'm still worried that if I don't do anything, email access will be terminated. Can I get something in writing that will be more convincing?

A copy of the email is below:

Hi <redacted>, We've recently made some changes to our email service and updated our
 email policy. We wanted to let you know how these changes impact the email
 service you receive from Orcon.  A more detailed view of our new email policy can be found in Section 6 of
 the Orcon General Terms and Conditions located at
http://www.orcon.net.nz/about/terms/category/general.  What are the important changes?  A couple of the more important changes to the Orcon email policy are the
 implementation of archiving and deleting of unused email accounts. Orcon
 will archive any inactive mailboxes after 6 months and delete unused
 mailboxes after 12 months.  We can see that the free email service on the Orcon username '<redacted>' has
 not been accessed in over 12 months.  If you do not wish to use this service, then simply take no action. On
 Monday 27th of October 2014 we will be cancelling the service and any
 email on this account will no longer be accessible.  If you still wish to use this service or have an email forward to another
 email address, then it is important that you log in, before the date given
 above, to the Orcon webmail service or use POP / IMAP from the main email
 address in your email application, at least once every 6 months to keep
 the account active. You can access the Orcon webmail from the main page of
 our website www.orcon.net.nz  For more information on setting up your Orcon email in an email
 application using POP or IMAP, please visit the Email section of our
 Support page on the Orcon website:  http://www.orcon.net.nz/support/wizard  Please call our Helpdesk on 0800 13 14 15 if you have any questions, or
 need assistance with anything. They would be happy to help.  Regards,
 The Team @ Orcon