Does student membership require permission from the school?

Who is paying for it -- you or your school?

If you pay for a personal All Apps plan and you're eligible for the student discount, your school does not need to know about it.  Use any e-mail address you wish.  But your school must be accredited.  And you have to submit proof of enrollment (student ID card, tuition statement, report card, etc...).

Carefully read the Terms.  If you don't understand them, ask a parent or teacher to help you.

https://www.adobe.com/creativecloud/plans.html#ccm-ste-introductory

What he said about schools owning the account if you use a school account

There have been several messages recently about problems with education subscriptions
Need IT Admin https://community.adobe.com/t5/get-started/creative-cloud-app-can-t-manage-apps-don-t-have-an-it-administrator/td-p/10608944

FAQ https://helpx.adobe.com/x-productkb/policy-pricing/education-faq.html

ID Proof http://www.adobe.com/store/au_edu/academic_id.html

If the school has an Adobe account, they won't just know, they will own it. If a school is registed (say mycollege.edu), then all the email addresses (like nora@mycollege.edu) automatically belong to the school in Adobe.  It isn't possible to have a personal account if the school has an Adobe account. If you want a plan outside the school, don't use a school email. (School emails have another problem too - when you leave the school you may lose the email, but keep paying with no way to cancel!)