In a medium saucepan, cover the potatoes with cold water. Add salt and bring to a boil over high heat. Cook until tender, about 15 minutes. Drain the potatoes.
In a medium saucepan of boiling salted water, cook the broccoli rabe until crisp-tender and still bright green, about 5 minutes. Drain well and cut into 2-inch pieces.
In a deep 12-inch skillet, combine the oil, anchovies and garlic. cook over high heat, mashing the anchovies, until fragrant, about 1 minute. Spread the mussels in the skillet, top with the potatoes, broccoli rabe and parsley. Add 1/2 cup of water and season with salt and pepper. Stir several times, then cover and cook until the mussels open and the vegetable are tender, abut 8 minutes. Discard any unopened mussels. Transfer the stew to soup plates and serve.