Of course, make sure your parents understand that skate boots are in some ways completely different than ordinary shoes.
A little bit of misfit in ordinary shoes usually doesn't matter much. What is more, since you have fairly wide contact between the bottom of the shoe and the ground, you can often make do with unequal pressures on different parts of the feet. Except in somewhat severe circumstances, the ground will force your shoes into approximately the right orientation. In fact, for many activities, you want a little bit of motion (like heel lift for hiking & backpacking) inside the boot.
But balancing on a fairly thin edge enormously increases the forces on your feet, and requires that you have very good control of boot & blade orientation. That in turn requires more more complete contact and close to equal pressure between your feet and almost all parts of the interior of the boot. If your current boots don't give you that kind of fit, and can't be modified to give it, or don't have the right degree of stiffness to help you maintain it, that could lead to very serious injury.
The fact that your foot is a sizeable fraction of an inch or more above the surface, just means the ice has extra leverage to tilt your blade more, that you need to be able to fight.
This control requirement is further emphasized by the aesthetic standards of figure skating. While rocking between pronation and supination is a normal part of walking and running locomotion (or speed skating), figure skating standards often require you to maintain a path with constant radius of curvature, which means you sometimes must maintain a fairly constant tilt. Which in turn increases the need for nearly full contact, and proper stiffness.
You can find various sources on the types of injury that figure skaters tend to suffer. I'm sure you will find mention on ankle sprains & strains, as well as foot bone fractures of various types. Of course almost all sports can cause injury, and foot & ankle injuries are among the most common, for a number of reasons. But the need to balance and control yourself across a thin edge, the elevation of the foot above the surface, and the need to create controlled pronation & supination makes appropriate boots much more important in figure skating than many sports.
But don't emphasize injury prevention so much that your parents decide you shouldn't figure skate. After all, I wouldn't be surprised if the injury rates are somewhat lower than other sports you might reasonably take up instead - e.g., you could mention a fascination with mixed martial arts (it's become somewhat popular training for people who enjoy certain video games), free climbing, hang gliding, tightrope walking, ballet, car racing, training for the military...
And there are other expensive sports you could take up instead. Horseback riding (you really need your own horse, at stable space & care to keep it in), parachute jumping, bicycle racing (race bikes now often cost as much as sports cars). It's all a matter of reasonable alternative activities that cost even more.