When you say you can pay for it- the appointment will be very expensive without insurance, probably $300 for a first visit, maybe more. If your parents have insurance, and you have the card then you can present and likely pay much less, perhaps just a $50 co-pay or less. In the United States anyway.

As to making the appointment, I'm sure that wouldn't be the problem, but in the US they will ask about insurance, so you will need that answer. Once you get there, you might be fine, and you might not, if you go alone- you are a minor after all.

If the issue is that your parents work and can't take off, what about an aunt or uncle or grandparent?

It depends on what state you live in (if you are American).

But the short answer is, probably not. Most state laws only allow minors to receive health care that relates to sexuality or drug addiction without parental consent.

For example, here is everything a minor can receive without parental consent in Colorado: