I'm currently in an online school. Let me give you a little advice about my typical day, and hopefully some insight and food for thought regarding your parents and your social interaction.
My typical school day starts at 6:30am. I get up, I get dressed, eat a breakfast, and set myself to work. I take six classes (credits toward graduation), and typically end my day at 1:30pm, taking exactly one hour per class and saving any uncompleted work for "home work" later that evening. From 1:30pm to some time after dinner (between 6:30pm-8:30pm) I complete chores, hang out with friends and work with my clients, all of whom use my dog-walking services. Being an introvert, I do not socialize with my peers but instead choose to hang out with friends that are many years my senior and often have odd jobs for me to complete rather than the desire to play video games.
Your social life in an online school is largely dependent upon your capability to balance the academic and the social aspects of your life. You have to create, manage, and stick to a schedule, while either maintaining friendships from your old school or finding ways to forge new friendships. Often times, getting a job is the easiest way to make friends. If the online school you choose has the option for clubs (music club, equine club and anime club are some options at my school) there's plenty of opportunity to meet and chat, albeit through text, with peers who have similar interests of you. Maintaining a social life is Not difficult with an online school, you just have to put forth the effort.
In regards to self-education, that's what online school is all about. In many ways it'd be similar to your current situation, the difference would be that you'd be completing your own work and your teacher wouldn't be in the same physical location as you. In my school, the curriculum that you currently sit at is tested with an initial exam, they place you in grades according to your results. During the first couple of weeks following the start of a semester, you can request a course change based on your opinion and whether you feel you're being challenged enough.
Every teacher is enthusiastic, at least they have been for my last two years. They want to see you succeed, and they want to help. It's your job to help them do theirs as much as it's their job to help you succeed.
Either way, if you're confident in your ability to succeed in an online school and maintain a social life, you should consider a different approach with your parents. Good luck.