Can my student use the parent's email for their eLearning course?

Yes, as long as you don’t have another student using it. Please be sure to read the "Student Email" section of this article: Self-Paced eLearning Subscriptions: Email Address, Parent Login, and Subscription Length

A filter can also be setup on the student's email account to automatically forward all emails from the eLearning course to the parent's email address. However, if you follow these steps to check your student's work, the notifications are not necessary: How to Check Student Work