Are either of them doctors? I'm not sure how this works in the states, but up here a doc could "suggest" you avoid certain activities for a certain period of time. If there's any chance of damaging your back I'd stay off the board until it's healed.
This coming from a guy who bruised his ribcage and is entering a competition tomorrow anyway!
As for not letting you board, that's a whole other issue between you and your parents! I was lucky enough to have parents that let me try pretty much everything and get my scrapes, as long as it was a healthy activity (instead of getting into drugs, drinking, etc.).
The one line I said to my mom in my teens, when she was being overbearing was "Mom, you did a good job raising me, now let me to my job (and live my life)!" She still quotes it. Show them you're mature enough to think for yourself... Not sure how you're going to do that, I'm not a parent and it's too long since I was a teenager! lol