You should consult a New Jersey attorney that is familiar with your state's employment and wage/hour laws. Your benefits are not covered by ERISA - the federal Employee Retirement Income Security Act - because you work for a local government entity (the school district). That means that state law will apply to your claim as it relates to benefits.
I would need to know more facts and do some research for you, but initially the key questions I have are: what is your agreement with the school district? are you treated as an independent contractor or an employee (you can tell this by the type of tax information you get - 1099; are taxes taken out of your pay by the school district)? Have you asked anyone at the school district about changing you to a full time employee?
If you are an independent contractor, it doesn't matter how many hours you work. They are not obligated to provide you with benefits.
If you are an employee, it depends on how they define "part time". But again, there are laws that prevent employers from taking advantage of part time employees, especially state employees (assuming you are working for a public school).
Again, I would need more facts to determine whether you fall under a certain category of worker and what the specific facts are of your employment to determine if the laws can help you get benefits, etc.