pocketbikes will let you "respect" big street bikes a little more, because crashing at 20mph on those hurts, and you can do the math for 60, 120mph etc.
You will learn countersteering, traction, mechanical upkeep and modifications, a little bike physics, all for about $200 capital cost on a cag.
I'd suggest a pocketbike, mid-bikes (types with electric and larger frames) are heavier, don't lean, and have the same engine in them ( = slower).
You can also have your full get-up of gear for a legitimate reason well before having the real deal. I'd suggest having a good set of GP gloves at the least with these bikes, rash your hands real good when wreck.
Dirtbikes are more dangerous than streetbikes in my experiences. WAY easier to gain false confidence, wreck, and usually while wearing less gear and in some nasty terrain. They definately help your skills, but it would be an oxymoron for your parents to let you ride dirt if they are worried about safety.