In a large pan of boiling salted water, cook the parsnips and swede, covered, for about 20 minutes until tender. Drain well, then mash them together using a masher or food processor until reasonably smooth but still with a bit of texture. Stir in the soured cream, horseradish and thyme and season with salt and pepper.
Spoon into a buttered shallow ovenproof dish and put to one side.
Make the topping. Melt the butter in a frying pan and cook the onion for 5-6 minutes, until gorgeously golden. Mix in the breadcrumbs and stir to brown and crisp a little. Season with salt and pepper and add the thyme. Take the pan off the heat. Spoon the mixture casually over the top of the mash. Scatter over the parmesan. (Can be made ahead to this point and kept covered in the fridge for up to a day. Or can be frozen for up to a month.)
Bake with the turkey (190C/gas 5/ fan 170C) for 35-40 minutes if doing from cold, 25-30 minutes if not, or 11⁄2-13⁄4 hours from frozen (put foil on top, and remove it for the last 10 minutes) – until golden and crisp on top. Serve scattered with a few more thyme sprigs and leaves.