Mmmm, I'm an Explorer Scout Leader, this "if I push him, will he break?" problem is a regular for us
As you are the elder brother you will be paving the way for your younger brother, so it's the first mile that counts.
It depends on how old you are to a certain extent but assuming you are 17-18 parents need to be shown that you are no longer 10 years old.
I'd suggest your first move would be to join up with a local cycle group for the summer, check out the CTC website, you can then go on a few of their group rides, a couple of 'family' 20 milers out to a pub will in the sticks which you tell your parents about (emphasizing the 10 year olds that came along) will give them a good idea that you can be responsible on the roads.
You need to prove you can do this with older 'responsible' people before they will let you do it on your own.
I've taken people as young as 14 half way across Europe on bikes (as well as into the Alps, and on hundreds of camps where they have to survive on their own)
So rather than starting arguments with the parents, join up with a cycling club and prove you can do it