I created a sub user on my premium account and they do not have access to the same amount of submissions

I created a sub user on my account and they still only have 100 submissions instead of 1000 like my account.

Sub account users are also the normal users just as they have individual account. However, these sub-users are allowed to share JotForms and submissions from your account so that they will be able to view your submissions/edit your JotForms as per the rights provided by you. If you want your sub-account users to have more submissions than a free user, that account is also needs to be upgraded. Please refer to the following guide that help you in sharing JotForms with sub-account user. 

http://www.jotform.com/help/232-How-to-Share-Forms-with-a-Sub-Account-User

Hope this information helps. Thank you!