I used my powersander on the bottom leading edge of the PVC pipe so that is pulls down as it goes through the water. I tested this today along the dock and the paravane now stays down in the water much better.
I also tried connecting the line directly to the side of the pipe instead of to two lines on the paravane. This works, even though I don't have the right spot yet. I just happened to have 3 holes on the pipe that were drilled in the wrong place. The right spot is a bit further back (maybe 2 or 3 inches) and a bit higher (maybe 1 or 2 inches). This will reduce the chance of lines getting twisted around something the wrong way. Probably easier to manufacture correctly too.
So maybe the Big Pipe Paravane is still a good way to go.