Heat oven to 450°. Place mussels in a large bowl of lightly salted cold water. Set aside.
Peel potatoes and slice into 3 x 1/2-inch sticks. Toss with olive oil, 1/4 teaspoon of the salt and the pepper. Scatter in a single layer on a baking sheet and bake at 450° for 17 minutes; flip potatoes and bake 15 more minutes or until they are browned. Remove from oven; toss with another 1/4 teaspoon of the salt while hot.
Meanwhile, debeard mussels by pulling off the seaweed-like string between the shells. Discard any open mussels. (If open, tap on the counter; if shell closes, you can keep it.) Place mussels in a new bowl.
Melt butter in a large, lidded pot over medium heat. Stir in shallots and sauté 3 minutes. Stir in garlic and sauté another 1 to 2 minutes. Add wine and remaining 1/4 teaspoon salt; bring to a simmer. Add mussels, cover and cook 6 to 8 minutes, or until shells have opened and mussels are cooked through. Stir in parsley. Discard any mussels that did not open.
Garnish fries with additional parsley, if desired, and serve alongside mussels and cooking liquid.