Tuva: what are your parent's primary objections? I ask because there could be many different answers to that question. I doubt that it is the first class education or free tuition that bothers them
Is it that your parents do not want you in the military because they are afraid for your safety? In that case, you may need to make it clear that you are planning to choose a military career with or without the Academy. You don't need their approval to join the military, and you would be wise to apply for ROTC in any case both to prove your commitment to the Academy and to have a "fall-back" if you don't get in. Once they see that your career decision, they may come to understand that the USNA experience is the best preparation for success you could possible get.
On the other hand, are they supportive of you serving in the military but fearful of the intensity of the Academy experience? In that case, you need to update their understanding of what life is like at the Academy these days. A visit to the Academy is important. Perhaps the most valuable thing would be to set up some interviews with Academy graduates. Your BGO can help with this. If you know someone who works on a local base, they could arrange a day to meet with officers -- most are delighted to spend some time with a prospective applicant. I had the opportunity through a friend who worked on a base to meet with several USNA graduates when my kid started down this path and it was incredible! Once I saw first hand what amazing people each and every one of these graduates was (and that they were still "normal" in all the good ways
I fully understood the draw of the Academy to my kid.
Finally, is their concern that the Academy experience puts limitations on their access to you, time with you, etc.? That is hard for parents but only at at first. As discussed on this board before, most of us will say we are far closer emotionally to our mids than we are to our kids who have gone to civilian college. See if you can get some interaction with parents of current mids or graduates. I'm sure this can be arranged through your state parent's club. And believe me, I worry more about my kids who are away at civilian college or living alone in a distant city than I do about my mid.
If you are proactive in taking steps like these hopefully they will see how motivated and serious you are. By proactive I mean: YOU make the phone calls, YOU set up the interviews, YOU lead them to understanding your motivation. If you're going to survive the Academy, you need to be ready to take full control of your life. My advice is to talk it through with them and then SHOW them that you are making this decision as the competent adult that they raised.
Good luck!