Cover the diced parsnips and potatoes in a saucepan with water and bring to a simmer with the lid on. Simmer for 25 minutes until the vegetables are soft. Pour off the water.
Process the vegetables into a creamy puree with the hand blender. Add a little hot water until the desired consistency is reached, then mix thoroughly with the remaining ingredients.
Heat the olive oil for the garlic oil in a small pan and lightly fry the garlic. The garlic should be slightly browned at the end, but of course not burn.
Arrange the parsnip puree on plates and arrange with the garlic oil, including the fried garlic slices, and serve.