Place a large fine-meshed sieve over a large bowl. Grate the potatoes and place in the sieve. Grate the parships and carrots and place in another large bowl. Add the onion, dill, matzo meal, salt, and pepper to the parsnip mixture.
Using your hands, squeeze the potatoes a palmful at a time over the sieve to remove as much moisture as possible. Add the squeezed potatoes to the parship mixture. When all the potatoes have been squeezed, remove the sieve and carefully pour off the potato liquid while leaving the white potato starch in the bottom. Add the starch to the vegetable mixture. Add 2 eggs and thoroughly combine. Put a scant 1/4 cup of the mixture into the palm of your hand and flatten with your other hand. If the mixture sticks together, you can form the remaining mixture into cakes. If not, add another egg and then form the cakes.
In a medium, heavy-bottomed skilled over medium heat, heat 1/2 inch vegetable oil. Place a few of the cakes in the hot oil and fry until dark gold on one side. Flip with a spatula and cook the second side until golden. Place on a baking sheet and keep warm in a 200 degree F oven while frying the remaining cakes. Serve hot.
Recipe courtesy of Chronicle Books, The Nantucket Holiday Table by Susan Simon.