I live in Seattle and turn 18 before the year's end, and I'm told my family has an income such that if I get a job and make more than $1000, then I would push my family into a higher tax bracket.

Because of this – according to my parents – our health insurance would no longer cover my braces and the family would all of a sudden have to pay $3000 to $4000 for my braces, which I would have to pay because this would be caused by my job.

I don't think that my salary at minimum wage would be enough to cover the insurance, given that I would probably only have the job over this summer. What options do I have?

Edit: The #1 reason I want a job is to have spending money.

Edit 2: We are in the upper-middle class.