Using Direct E-mail to send to a contact requires at least read access on the Parent Customer record.
Quite why this is needed is a mystery. Possibly a bug.
[13th October 2014] I might have discovered the reason for the above. If you select two contacts, where one does not have an email address, Send Direct Email will send an email to the contact that does have an email address and an email to the parent of the contact that does not have an email address (provided that the parent has an email address). That discovery surprised me and I cannot find any documentation that describes this functionality or how to stop it. I tested this in CRM 2011 and CRM 2013.