Squeeze out any moisture from the grated potatoes and parsnips. Mix together in a bowl and season with salt and pepper. Heat the butter and 1 tablespoon of the oil in a large nonstick frying pan. When the butter is melted and foaming, add the potato mixture and spread out evenly. Cook over a medium heat for about 5 minutes until the base is golden and crisp.
Place a large plate or baking sheet over the pan and turn the rosti out. Slide it back into the pan and cook for about 10 minutes until the potato and parsnip are tender and golden.
Meanwhile, heat the remaining oil in a separate frying pan. Add the bacon and mushrooms and fry for 5 minutes until the bacon is crisp and the mushrooms are tender.
Cut the rosti cake into wedges and serve with the bacon, mushrooms, baked beans and ketchup.