Let me try and clarify this one.
Actual doctor confidentiality and parents looking at an insurance statement are very different things.
Understand that for starters, in the US, many youth don't get sexual healthcare via their parents insurance, but instead through private clinic systems.
And no, if a parent were to call doctors or clinics, unless a doctor believed that parents need to be apprised of a serious health or safety issue, it would be unethical -- though not always illegal -- for a doctor to report to a parent. Some situations, such as being under the age of consent, can inluence that. That's not necessarily simply 16 or 18.
But yes, if a doctor wasn't respecting pateinet confidentiality overall, a given clinic or practice would likely not keep he or she on.