Preheat oven to 180°C. Place oven rack in lowest position.
In large roasting pan or 2 smaller pans, combine potatoes, parsnip, celeriac, onion, thyme, salt, pepper and oil. Place spatchcock, breast-side up, in among the vegetables; sprinkle with paprika.
Bake for 50 minutes, stirring vegetables occasionally. Stir honey into the vegetables. Bake until spatchcock are cooked through and vegetables are fork-tender, 10 to 15 more minutes. Cut each spatchcock in half to serve.