Well if I know anything about parents, its that it is very difficult to change their minds on something they deem unsafe. All I can say is that city bikes probably are more probably more safe in the aspect of having better control, but I am sure you can get good enough on a road bike with practice. The most important thing to note is that even if you are the best rider, no matter what bike you have- vehicle drivers can be very unpredictable and can hit even the most seasoned cyclist.