Can my employer legally do this? (see details)

I was hired as a part time employee. I let them know before I was hired that I would like to keep my hours to a minimum (around 20 hours a week). Next week I’m on the schedule for 35 hours. Is this still considered part time? That’s almost 40 hours! I have a feeling that they’re trying to get me as close to 40 as possible so that they don’t have to give me full time benefits (health insurance, life insurance, vacation time, etc). Is this legal? It’s very frustrating. If I wanted to work 35 hours a week I would have asked for a full time position!

Note: I live in NY state, if that matters.