If you are under 18 and your parents say no, the military will not try to undermine your parents beliefs. On the other hand, if you are over 18 having your parents disagree with your career choice is nothing more than a "family" problem. My suggestion would be for you to draw out "why" your mother/parents are so opposed to USNA/military (often times the unknown is what puts them off). During a BGO interview things like this can come out when speaking with the candidate and their parents. Your parents negativity toward the SA/military will NOT impact your chances of receiving an offer of appointment. We would note that mom/dad are opposed and at the same time make mention of why YOU, the candidate, is making the choice to join the military/SA. Take a breath, don't get too anxious about this and take this opportunity to open up an adult conversation with your parents about your future plans. PM me if you have any other questions/concerns.
Good Luck.
A couple thoughts ...
1. While I'm not sure what is meant, but my own read is that yours is "nothing more than a 'family problem' " risks portraying your situation as age-dependent, and/or less signficant than it likely is. Imo, this has nothing to do w/ age. It is about you and your mother and your relationship, which has nothing to do with whether you are 17 or 18. This is not an issue of legalisms, and attending a SA beyond being the source of consternation and disagreement could become a very disruptive, corrosive issue for your life. By your continuous concern expressed on this forum, you obviously have deep care and respect for your mother. That is a very postitive starting point. Go from there.
2. While some BGO's might disregard discovering that a parent(s) are either unsupportive and/or outright hostile toward their child pursuing a SA appointment, most would not. And I'd offer my own opinion that any who would disregard are being irresponsible in their work. Always, this will impact upon a Mid's experience, one way and usually in many. And I cannot imagine that USNA knowing there is this significant issue, would disregard it in considering offer of appointment. Why would they set the institution up and you for potential failure? That one makes no sense whatsoever, and as we hear time and time again ... inversely ... the #1 reason Mids bail out? Ma and/or Pa, not the Mid, wanted them there. And it works the other way. Ma and/or Pa did NOT want them there. Especially females.
I would concur as was suggested, an appropriate adult 3rd party might be able to assist in reconciling this very significant issue between you and your parents. A pastor, priest, rabbi, spiritual leader, counselor, respected family friend/neighbor, someone respected by both you and your folks.
Last thought on this ... and I'll say it simply. Do not disregard your parents will in this. It could become a very destructive thing in doing so.
Blessings and best wishes in working this out. But your answer to this one is not on this forum. You risk greater confusion as you might consider mine or others counsel and perhaps misguided desire to help you in this. I seriously doubt it.