It seems like you are stuck between a rock and a hard place. I really don't envy your position. It seems obvious that you are certain of your decision, and you really should do whatever is possible to achieve your goals. Before you leave, you need to consider a few things. First, are you prepared to go through your training and deployment on your own with little to no support from home? That is VERY difficult. I have seen many a college student get homesick, feel the pressure, stress and try to deal with the responsibilty of being away from home and literally crack under the pressure.....and that's just college. This is1,000 times bigger than that, and you may not think you need it, but having your parents support you through this would be huge.
Also, joining the armed forces is and should be about honor and respect. Please please please honor and respect your parents by telling them your plans. You do not need to leave now to get your orders, the Navy will be happy to take you at any time. There will be another class after you graduate. Your recruiter probably has a quota he has to meet and will push to get people through. Remember that.
Lastly, if you go to college and get a degree, you can enter the Navy as an officer and get better pay and benefits right from the start, and they probably will still payoff your student loans and still give you a signing bonus.
Your choice to join is not wrong at all. Just make sure you are already making the mature decisions a seaman will need to make.
Just my opinion......