Put the potatoes in a large microwave-safe bowl and cover with water by about 1/4 inch. Cover with plastic wrap and microwave until just tender, about 6 minutes; drain.
Meanwhile, heat the olive oil in a large Dutch oven or wide pot over medium-high heat. Add the onion and garlic and cook until soft and golden brown, 5 to 6 minutes. Add the potatoes, paprika, allspice, cayenne and 1 1/2 teaspoons salt; stir until the potatoes are coated with the spices and begin to soften, 2 to 3 minutes. Stir in 1 cup water and the tomatoes, scraping up any browned bits from the bottom of the pot. Bring to a simmer, cover and cook until the potatoes are tender, about 10 minutes.
Stir in the mussels, olives and parsley. Cover and cook until the mussels open, 4 to 5 minutes. (Discard any that do not open.)