It's good to have parents who care, but I also remember how it could feel a bit stifling at times. My parents used to worry about me bicycling everywhere when I was younger. By the time I was 13, I was regularly going for 10 or 15 mile rides...although I had to promise to stick to the quiet streets. I had to clear every route with them, and I also wore a helmet (which was not very common back in 1983). When I was in high school I raced and regularly rode on busy streets and country highways. My parents definitely worried then. Ironically, the only time I was ever hit by a car was when I was bicycling through a quiet neighborhood with very little traffic. I got hit by some idiot who decided it would be fun to take a curve in the wrong lane (which happened to be my lane) and he hit me head on going pretty fast. Amazingly enough I was not seriously injured (although my bicycle was totaled). To say that made my parents worry even more is an understatement, but by that time they realized that bicycling was something I just had to do... and they were very supportive in helping me get back on the bike as soon as possible.
Tell them you appreciate their concern, and assure them that you know how to ride safely (stay visible, be predictable, and ride defensively... but signal and take the lane when you need to). It might help if you could find someone else who bikes that same route. Maybe you could ride with them for awhile?
Sean