My parents were dead set against my joining the profession, as well. Further, they didn't have any money to withhold.
So I took control of my life the old-fashioned way. I put my right hand in the air, got a silly buzz cut, and learned how to march. Uncle Sam paid for my college, taught me to fly, then paid me to accrue enough hours to get a job with a legacy carrier.
I've flown with enough Asian-American pilots that I was surprised to hear about your parents' racially-based concerns. Those concerns have no basis in fact. (btw, I've trained pilots who are five feet tall with no problem at all.)
Here's how I'd play it, if I were you: take their money and get that degree in engineering or accounting. As another poster said, you need a fallback to survive furloughs, layoffs, depressions, etc. Then go to one of the civilian flight academies or officer's candidate school and live your dream.