As a private organization, it is not illegal under federal law. It may, however, be illegal under the laws of your state.
For tax purposes, the IRS uses common law. Their language:
A worker is an employee if the firm has the right to control what will be done and how it will be done.
It is unlikely as a cook you control what will be done and how.
From the SS-8 form and the facts you have provided, it appears you qualify and should be a W-2 employee. Your status as part-time employee is not a factor.
For more information and a copy of the form, follow these links:
Form SS-8 Instructions
Form SS-8
