Roll your entire foot? I'm not sure what that means as I was never in marching band, but the pain is probably from your heel hitting the ground or "too much too soon". Your feet probably are lacking a lot of muscles from being in shoes for years. Try to build up foot strength by being barefoot/minimalist more.
The marching step does involve a heel strike, followed by a gentle roll through the rest of the foot. However, it's intended to be a relatively gentle strike. Marching shoes have little padding, and if the heel strike is sharp, it will interrupt the flow of air, causing problems for wind instruments (woodwind and brass). It is a heel strike, but if done right, it's a gentle, controlled heel strike. I'd think pain wouldn't be from marching in them, but if you're still walking with a heel strike, that could be related. I alternate between a heel strike and a midfoot/forefoot strike when walking in my VFFs.
In terms of your parent's approval - It's a challenging topic. You have to convince them that they are for more than just trend following - they serve a purpose. You could then explain their benefits - do some research.
Essentially though they act as a barefoot replacement with a little padding. They allow you to strengthen your natural muscles instead of relying on your shoes. Plus it's good for your back, knees, etc. That's the argument I have/would use with friends/family.
Honestly, instead of trying to convince your parents the shoes are beneficial (which is something that doesn't have solid proof), convince them they're more comfortable and better liked by you. People who question the purpose of the shoes tend to respond very well when I say something like, "Well, I'd be barefoot if it wasn't unsafe and illegal, in some places." These shoes feel very close to being barefoot, and that's a fact, not a debated hypothesis. Coming with hard facts is more likely to do better than coming with theories, even if they're good theories with good backing.
Then again, my parents were the sort who believed every rule should have a purpose and the purpose should be explained. If I could logically justify something, I could do it. Coming to them with a, "It's better for me," would have needed significant proof, but, "I like them better and will use them more, and more willingly," would get me far.