I feel alone and I'm only 13? :(?

[Megapurrr]

If you are 13, from the way your wrote I gather that you are a rather bright young guy/girl. Although this will be VERY hard for you, I think that you should, regardless of all the screaming, confront your parents on the topic.
No child should grow up with no social life, not just because being friendless is no fun, but because it is important to be around people to develop the social skills you will need to have.
Tell this to your parents. Tell them that if they are so insistent in you being home-schooled, that they need to find another way for you to meet people.
Ask them if your happiness means anything to them, and if so, to let you have friends. Explain that facebook is not the devil. That it is the equivalent of one big billboard where people can write things. That facebook itself filters content and does not allow for any inappropriate images.
Tell them that you NEED this, because you are starting to feel lonely and depressed.
Tell them that, as much as you love them as parents, that you need to talk to people your age. That you need to live your own life, otherwise when you become an adult, you won't know how.

And if all else fails, do as that other guy said. Start talking with Ernie, your imaginary friend. That should freak your parents out enough to do something about it all. If that doesn't work, start doing moderately mischievous things, and blame them on Ernie, or tell them that Ernie told you to do it.
If they take you to a shrink because of that, tell the shrink the truth about why you are doing it right away. He will then convince your parents to let you have a social life.

Best of luck buddy. I really hope things change for you.
Always remember, it's all temporary. As soon as you start working, or going to collage, you'll meet tons of people.

Be aware though, the social life of a public (or private) school child is not always so glamorous and great. It is especially hard to make friends when you join a school later, as all the social groups will already be formed, and it will be really hard for you to integrate at first. You will integrate, it will just take time and work. Don't be discouraged.


Edit*:

And don't listen to the people that are telling you that you don't know real loneliness. Just because you are young, doesn't mean you can't be lonely. They are speaking out of their as*es. I know what losing loved ones is, and what break-ups feel like, and I also know that a 13 year old can feel just as bad from loneliness. They don't know what they are talking about. What you are feeling is real.

Also, don't listen to the ones that are telling you that you will have plenty of time to make friends later. Don't get me wrong, you will have time later, but you have to live life to the fullest. You don't know if you'll even be alive in 10 years. It is important to plan ahead, but it is also important to enjoy your life. You are only young once, so keep a healthy balance of enjoyment and planning.
Also, if you never learn to make friends, you won't have the ability to make friends in when you are older. And believe me, that is one skill that is MUCH MUCH harder to learn as an adult.