In a 6- to 8-quart pan over high heat, bring 3 quarts water to a boil. Add beans and cook until crisp-tender, 1 to 2 minutes. Remove beans from water with a slotted spoon or strainer and rinse under cold running water until cool. Return water to a boil. Add potatoes, reduce heat as necessary to maintain a simmer, and cook until potatoes are tender when pierced, 15 to 20 minutes. Drain.
Meanwhile, in a small bowl, mix olive oil and brown sugar. Place onion slices and sausages on a baking sheet and brush generously with brown sugar mixture. Broil 6 inches from heat until onions and sausages are lightly browned and onions are tender when pierced, 8 to 10 minutes. Let stand until cool enough to handle. Coarsely chop onion. Slice sausages 1/2 inch thick.
In a large bowl, combine beans, potatoes, onion, sausages, mayonnaise, mustard, vinegar, olives, and parsley. Mix gently to coat, adding salt and pepper to taste. Serve warm, or chill until cold, about 2 hours.