Center a rack in the oven and preheat it to 325 degrees F. (If your oven can't hold three 9-inch cake pans on one rack, position the racks to divide the oven into thirds). Butter three 9-inch round cake pans, dust the interiors with flour, and tap out the excess, or use baker's spray.

Whisk the flour, coriander, baking powder, baking soda, and salt together. Put two teaspoons of the sugar in a small bowl and stir in the minced ginger and zest.

Working in a stand mixer fitted with a paddle attachment, or in a large bowl with a hand mixer, beat the oil, the remaining 1 cup sugar, and the brown sugar together on medium speed for about two minutes. (It's fine if the mixture looks grainy.) One by one, beat in the eggs and then continue to beat until the mixture is smooth and velvety. Beat in the vanilla, followed by the ginger-zest mixture and any syrup that might be in the bowl. Turn off the mixer and add the flour mixture all at once.

Pulse the mixture to start incorporating the flour, then mix on low just until the dry ingredients almost disappear. Add the parsnips and nuts and mix to incorporate. Switch to a flexible spatula and gently fold in the cranberries. Divide the batter evenly among the three pans and smooth the tops.

Bake for 33 to 37 minutes, until the cakes are golden and just starting to pull away from the sides of the pans. The tops will feel springy to the touch and a tester inserted in the center will come out clean. If you're baking on two racks or your oven has hot spots, rotate the pans from front to back and top to bottom after 18 minutes. Transfer the cakes to racks and cool for five minutes, then run a table knife around the sides of the pans and turn the cakes out into racks to cool to room temperature. (You can make the cakes a day ahead - wrap them well and keep them at room temperature.)

Put all of the filling ingredients in a medium saucepan, stir, and cook over medium heat, stirring, until the mixture bubbles, many of the cranberries pop, and the sauce starts to thicken, about ten minutes. The filling will thicken more as it cools. Scrape the filling into a bowl and cool to room temperature, then cover and refrigerate if you're not using it immediately. (It can be made up to three days ahead and kept in the fridge.)

In a large mixing bowl with an electric mixer, beat the cream cheese, butter, powdered sugar, and salt together on medium speed until very smooth; scrape the beater and bowl down frequently. Add the vanilla and blend.

If the tops of the cakes have mounded, you can slice away the crowns to even them. Take the first layer and generously cover it with frosting. Spoon half of the cranberry filling into the center of the frosting and spread it so that it comes to about an inch or two shy of the edges of the cake. Place the second layer on the cake, top side down. Cover it with frosting and spread the remaining filling over it. Cover the top layer with frosting in a swirling pattern. If frosting is left over, frost the sides of the cake.

Make a simple syrup by boiling the sugar and water together, stirring, for three minutes. Remove from heat, drop in the cranberries, and roll them around to coat them with syrup. Lift them out with a slotted spoon and transfer them to a rack. Let them set for one hour until sticky and tacky. Roll them around in the additional cup of sugar and let them dry on a clean rack for another hour. Place them on top of the cake when just ready to serve (they will not last in the refrigerator).