Here's a source that says you can successfully start parsnips in containers and transplant them. Here's another source that says they can be transplanted, but that it's difficult to do successfully. Note that it says difficult, not impossible.
Many plants that people recommend against transplanting, actually can be started in cells and transplanted if you're careful. I have started corn (maize) in cells and transplanted it, even though basically no one does that, and I found references saying that they wouldn't transplant well. I had an excellent corn crop that year.
Since you know that sowing turnips directly into the ground won't work in your plot, it makes sense to try starting them in containers. When you transplant them, be particularly gentle with the roots. (It is a root crop after all, so you don't want to damage the tap root.) The decomposable cardboard pots sounds like a good idea, since you won't be disturbing the roots when you transplant.
It occurs to me that mice sometimes like to chew up cardboard to make their nests out of, so there's a risk that they chew on the cardboard pots and disturb the plants. If there's any extra cardboard sticking up above ground level when you transplant plant them, it might be a good idea to trim that off.