Hi Michael,
Your beds sound productive and I am sure that 16 inches is plenty of depth, even for parsnips. I have grown lovely 6 inch carrots in 6 inches of compost on top of rotting grass. Also as well as the visible, edible large part of carrot, they send finer roots down much further into whatever the lower soil is, in your case, the clay (probably similar to here). I am always amazed when I manage to extract the lower reaches of my parsnip roots and see the yellow clay they have burrowed through.
I wonder whether your linkabord planks are strong enough to take the weight of the beds’ ingredients or do they need an extra support stake into the ground, along the sides?