From the studies I've seen, a vest reduces the risk of back injury by a factor of 5. However, IIRC, it doesn't decrease the chance of a broken neck or back, just the chance of injury to the rest of the back.
I don't jump, but my first fall from a horse was when my mare jumped out from under me (spin, leap, bolt) while I was dismounting in Jan 2009. My back is still tender every day in the area where I landed on a couple of rocks. Just because it is 'soft tissue' doesn't mean it doesn't hurt, or that it will heal. I've now gone 26 months since the fall, and it still hurts daily.
Listen to your Dad.
Besides, I'm a Dad with 3 kids and 3 grandkids, so I'm all in favor of listening to Dads...