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Preheat the oven to 180C/gas 4/fan 160C and butter a shallow ovenproof dish. Layer the potatoes, then the parsnip in the bottom of the dish with two of the sliced garlic cloves and sprinkle over a little salt and half the rosemary. It’s best to keep the potatoes buried under the other vegetables because they go black if they are on top of the gratin.
Pour the cream and milk into a large saucepan, add the rest of the rosemary and garlic, the carrots and a little seasoning. Bring to the boil, then turn the heat to the lowest it will go and simmer for 3 minutes.
Pour the carrots and cream over the other vegetables, and spread the carrots out. Sprinkle over the parmesan, cover with foil and bake for 40 minutes. Remove the foil and bake a further 20 minutes until golden and bubbly.