Preheat oven to 420ºF. Toss parsnip strips with 2 teaspoons olive oil and place on a parchment or foil-lined baking sheet. Bake for 15 minutes, then turn and bake until soft and golden, about another 10 minutes.
Saute sliced onions in 1 teaspoon olive oil over medium heat until nicely caramelized and soft, about 10 minutes. Set aside.
In a medium bowl, mash black beans with a fork, leaving some whole pieces. Add 1/4 cup bulgur wheat, chopped onion, garlic, cumin, salt, and hot sauce. Stir together until well combined and, using hands, form 3 equal round patties.
Heat remaining tablespoon of olive oil in a medium skillet over medium-high heat. Add black bean patties and cook about 3 minutes each side, or until golden and crispy on the surfaces and heated through. Remove burgers from skillet and set aside. Add 1 teaspoon olive oil and saute spinach until wilted, about 30 seconds to 1 minute more.
Serve one black bean patty on top of sauteed spinach, topped with caramelized onion and drizzled or dipped in mayo. Serve 2/3 of the parsnip fries on the side. Drizzle with additional hot sauce if desired.