Consumer questions are always welcome here. You just might have to deal with a few cynical remarks in the answer
I can't speak to New York auto policies, I write in California. In my state (and I assume this applies in New York, but I don't know), a covered driver of a car does not include anyone who lives in the house, unless they are listed on the policy.
A couple of examples:
When you are away at college, and do not reside in the home on a regular basis, when you visit your parents for a weekend (or spring break), you would be covered under permissive driver rules. You are not a normal or frequent driver of the car and would not have to be listed.
On the other hand, when you come home for the summer, or come home where you will be driving the car on any sort of regular basis, if you are not listed but then end up in an accident, your policy may not pay to cover the accident.
There are a lot of reasons behind this, but in general, every driver adds risk to the insurance company. This added risk is usually acceptable to the insurace company, but they may want to have some additional premium to cover the added risk.
Again, I'm not trying to be a New York agent, I'm responding with California answers, which may or may not cross state lines.....
Many carriers have a way to deal with younger drivers who are frequent but not regular drivers of the vehicle. This is especially true if they are still registered in school. They will have a classification such as 'away at school', which acknowledges the use of the car, but that it is with more regularity than a permissive user, but not as often as someone who lives at home full time.
They may also have a 'part-time' driver classification, which basically describes the situation you mention, driving infrequently. The primary driver is the parents. This does not remove any additional premium, but may reduce it. There are always a lot of variables in this type of situation though.
Dan