Peel both the potatoes and parsnips. Then slice the potato finely using a sharp knife or a mandolin slicer if you have one. Do the same with the parsnips.
Butter a large ceramic baking dish. Sprinkle with pepper and salt. Then top with a layer of sliced potato (about 1 potato). Season lightly with pepper and salt. Cover with a layer of sliced parsnip. Season lightly again with pepper and salt. Sprinkle with some of the chopped garlic and add little knobs of butter too.
Then keep making layers of potatoes and parsnip like this until you end with a layer of potatoes. Season with pepper and salt. In a small saucepan pour the cream and milk. Also add the strong mustard as well. Season with pepper and salt.
Cover with a piece of tinfoil. Put the potato dish in a preheated oven at 428°F (220°C) for 1 hour until soft. Remove the dish and the tinfoil. Then divide the fresh thyme and the rest of the butter over the potatoes.