Preheat oven to 350°F. Grease a 9-by-13-inch baking dish and set aside. Melt butter in a nonstick skillet over medium-low heat. When foaming subsides, add garlic and sauté until light golden, about 2 minutes. Stir in mustard. Add bread crumbs and sauté until they are golden and crisp, about 5 minutes. Remove from heat, season with salt and pepper and let cool.
Place broccoli in prepared baking dish. Pour heavy cream and 1/2 cup Parmesan over broccoli and mix well. Season lightly with salt and pepper. Mix remaining 1/2 cup Parmesan and parsley with cooled bread-crumb mixture. Sprinkle mixture evenly over broccoli and cover baking dish with foil.
Bake casserole for 20 minutes. Remove foil and continue baking 15 minutes longer, until casserole starts to bubble at corners, and crumbs are crispy