Should I move back to my home country?Go for it - why not. You already know you like your home country and you're already independent of a physical location due to the online nature of your business.
This is the ideal model that wantrepreneurs aspire to... not be tied to a boring-ass yet highly stressful 8-6 corporate job, work from anywhere, be it a coffee shop or even another country.
If you can keep your online business going from anywhere, go for it. The way we do things and 'work' has changed drastically since our parents' generation.
It is sometimes hard for parents to understand, firstly, what the hell we kids do every day (especially if it's got anything to do with IT) and secondly, why we don't do things the same way they did.
In our parents' generation it wasn't uncommon to be a 'lifer' at a company - once you got a job with a reputable company, you stayed there until your golden years.
Now it's not uncommon or even frowned upon to be switching not only jobs but also industry every couple of years.
The "American Dream" is a hugely successful marketing ploy.. sure, yes, it's possible to make it big, have your own business, make good money... so why not make it the "New American Dream" - you can run your online American business from your home country and still "dream American".
You need to do what's right for you. Your parents will understand eventually and support you. What's the worst thing that could happen? You suddenly hate it in your home country? Well.. then you just swallow your pride, tuck your tail in between your legs and come back and deal with smug/"I told you so" parents.
Worse things have happened.
Parents think that if you are secure (i.e. more money) then you will be happy. It's not that they don't want you to be happy. . they just think that the way to happiness and their happiness is to know you are taken care of and secure.. . in their eyes it's a win win.
You need to get to a point where you don't need the validation of your parents. . .that means you have grown up.
It's your life.