What are your rights as an part time employee?
Your question is too broad. Your rights depend on numerous factors: what type of rights you are referring to, the federal and state labor laws, any employment contract or any union contracts.
Being a freelancer means you run your own business, you spend for your own equipment, you don't have employment benefits, and you will need to track your invoices for tax reasons. Being a part-time worker means you are still employed by a company, only that you are just working lesser hours than a full-time employee. You may have options to choose your fringe benefits as a part-time employee.