No, your parents won't need a license. And you don't need an R1. There are good reasons the insurance on such a bike is as high as it is. A significant number of wrecks happen on such large bikes, especially by inexperienced riders...people such as you. Even if your parents register the bike in their name and are the primary policy holder, I'd expect the insurance to still be fairly high because YOU have to be added to the policy as a person who will be using the motorcycle. But you will be saving some money because your parents might get a multiple-vehicle discount, if the insurance is done through the same company as their car(s).
I once received a motorcycle insurance quote WITHOUT having had a license at the time. But my quote was probably higher, too. You only need a license, if you're driving it on public roads, not for insurance purposes.
The R1 might've felt "right" based on the limited experience you had with the bike, but that doesn't guarantee you can really handle it.