Should I apply for a master in the US? I am a CS student in Italy, about to get my bachelor degree. In the past few months I've been looking for some masters in my field (Computer Science with orientation towards Data Science) and have found some really interesting programs in top universities (some are Ivy) in the US.
I would really like to apply. I think I have the right stuff to get in, of course I'm aware of the low acceptance rates but I'm willing to try and give my best.
The only problem is my financial situation. I'll be considered as an international students and I didn't see many available scholarships; I'll just have very few possibilities to get private loans. My parents don't want me to apply because of the fact that if anything happens (health wise, for example) and I'm not able to complete my master I will be in debt of hundreds of thousands of dollars and with no equivalently repaying job.
Given my situation, should I pursue this dream of mine or try for a less expensive university in Europe?   

EDIT: The real question would be: Is it worth it to apply? Will it be worth it to be in debt of a huge amount of dollars to be in one of the best schools in the world? Considering that college applications are pretty stressful and obviously require time investment, I don't want to pursue a process like this and back down at the end (in case I get in). Is this a once in a lifetime chance or would I get a similar excellence in education in good European universities?
First of all I think you should apply and then decide, based on where you are accepted.

If you can find equal level masters in Europe (I'm sure there are) that cost less, can maybe guarantee you a scholarship, and are overall less risky from a financial perspective  you should aim to these ones. Consider especially German speaking countries, like Germany itself or part of Switzerland, that offer masters in English, in this way you'd have to learn Germany which is probably the second most important language to speak in Europe. This  could open a lot of chances in those countries after you graduate.

If for some reason you only want the USA you have to find out first if it's doable from any practical viewpoint, how much your parent disapproving is gonna affect you in every possible way (mental, economical...) and considering these factors how much you want it compared to the risks. 

Answer to your edit: 

There are excellence in Europe, you could talk to your professors and ask them about good places.
If you want to go outside of Italy the closest place is Switzerland, with ETH and UZH in Zurich, Lausanne polytecnique and the center for computational science at the Italian Switzerland university in Lugano being among the most famous and excellent places there. Lugano may be particularly good economically and practically since you could live in Italy (Italian cost of life) and cross the border to go to lectures. They speak Italian there so you won't learn another language, but it may be more comfortable. Going farther you can look anywhere for great universities. Take Germany, UK and Netherlands for example, they surely have great universities in your field.