How to Paddle SUP Over Surf
Getting over white water (surf) basically has two techniques: one with the parallel stance and one with a surfing stance. For the parallel stance I only recommend in very small white waters where paddling and balance will not be much of an issue for you.
First, the parallel stance. You'll want to take at least three to four good strokes as you approach the white water so that you have some speed to punch through, with your last stroke being just before the white water hits you so that you can drive through it and also have your paddle in position that allows you to use it as a balance stabilizer.
In a surfing stance going over white water, you'll start out in a parallel stance, get as much speed generated as you can approaching the white water. At the last moment, you'll take a step back with your rear foot. On your last stroke, place it deep just in front of the white water before it hits you so that you can continue to pull as you're going over the white water, and then the paddle will also be in a position to help you stabilize yourself once you're on top of the white water.
I recommend as quickly as possible once you're over, getting yourself back into a parallel stance to help deal with the balance issues that may be presented by an uneven white water.