Heat the oven to 325 F. Generously butter an 8 x 5 inch loaf pan. Add a few spoonfuls of flour, shake to coat all the surfaces, and tap out any excess.
Stir together the lemon juice and powdered sugar in a bowl. Stir in 1 teaspoon of the lemon zest and several twists of pepper. Set the glaze aside.
Put the parsnips in a food processor and process until they are finely chopped, like the texture of couscous; you want to have about 11/2 cups. Or you can grate them. Return the parsnips to the processor if you've taken them out to measure
Whisk together the flour, baking soda and salt in a medium bowl. Add the dates and toss until the dates are coated with flour so they don't clump together.
To the food processor, add the eggs, granulated sugar, brown sugar, vanilla and remaining 1 teaspoon of lemon zest and pulse until smooth and well blended. Pulse in the oil to make a creamy consistency. Pour the parsnip mixture into the dry ingredients and fold until blended. Ad the hazelnuts and fold until there are no more bits of flour.
Pour the batter into the prepared pan. Bake until slightly risen and a pick or thin knife blade inserted into the centre comes out clean, 35 to 45 minutes for a metal pan, longer for a glass or ceramic.
Let the cake cool in the pan for 10 minutes, then run a knife around the edges to release it from the pan. Set the warm cake on a rack set over a tray to catch the glaze. Drizzle the glaze all over the top while the cake is still warm.