It sounds like they're more concerned that you will suffer academically for having a relationship than anything else. It's probably a reasonable worry - relationships can be stressful and complicated. I think it may fall to you to convince them that you will continue to focus on your studies rather than allowing a relationship to divert you from achieving potential.
I know a fair few people whose parents are concerned that meeting someone will detract from their academic performance (and fair enough too - parents typically want the best for their children and academic success is probably the best way to improve your life chances). I don't think there's a simple solution to this, aside from reassuring them that you will continue to give study a priority. As for whether you actually do, that falls entirely to you.
Unless I'm misreading the situation entirely, I strongly doubt that they would cut you off for having a relationship that they didn't approve of. I know my mother would never have done so, no matter how seriously I disagreed with her. In general, I believe it's customary for parents to care about their children, and you'd need to do something pretty bloody heinous to offend them enough to cause them to take such extreme action.
As for the personal aspects of the other half of the relationship... well, again, their worry is probably legitimate but as has been observed many a time, it's not their decision to make. I suppose the crucial thing is to convince them that depression etc. will not have a damaging effect on the other half. Depression is a difficult one to get a handle on, as different people react to it in different ways.
Best of luck with it all. I should imagine that the parents will become more accepting with time (particularly after college), and I hope it works out.