Every type of education at home requires parental involvement for the student to be successful. Online school of any type will require the least. However, by high school even independently homeschooled students often learn independently. I'm sorry to say this, but no parent is an expert in math, foreign language, science, history and writing. Etc. Neither is any teacher for that matter. That's why independent homeschooled students don't always learn exclusively at the kitchen table. Instead, they take independent single online courses. They take Co-op classes. Only the most immature or the most over-controlled will need Mom to hover over them. Most curriculum at that level is self-taught for that reason. Parents oversee it, they don't lecture. However, in any at home situation, if you don't have parental support you're not going to be successful.
Online Public Schools require your parent to agree to be your learning coach. That does not mean that your parent needs to sit over you. That means that he or she needs to be involved in your education being sure you're getting your work done. An online private school might not, but still if you expect to be successful, don't expect your parents to be checked out.
By the way, in some states but not all, online private school falls under independent homeschooling laws. So if you're in one of those states, don't forget to comply with any registration or notification because some "expert" on Yahoo answers thinks you're not part of her exclusive club of homeschooled students. If you choose to do any kind of schooling at home understand the laws that apply to all types of education at home in your state.