I'm going to guess your hybrid bicycle has flat bars and the road bike has drop bars. Try taking the hybrid out for an extended ride (maybe a few extended rides) and see how your back feels after that. Then look at the handlebar height variation between the two bicycles. Maybe you need to raise the bars (or go with flat bars) on your road bike. I did that (raised the handlebars on my road bike a little bit with an adjustable stem) and my back issues went away.