1. In a pot with a steamer basket, steam the potatoes until tender, 10 to 15 minutes.
2. Meanwhile, rub the inside of a large bowl with the cut sides of the garlic clove. Set garlic aside; add the vinegar, mustard, capers, scallions and 1/4 teaspoon of the salt to the bowl. Whisk in 4 tablespoons of the oil. Transfer 2 tablespoons of the dressing to a small bowl.
3. Add potatoes to the large bowl and toss with dressing. Put the green beans in the steamer basket and cook until crisp-tender, 5 to 7 minutes. Add to potatoes and toss.
4. Season the salmon fillets with remaining salt and a generous amount of black pepper. In a medium nonstick skillet, warm remaining oil over high heat. Add garlic halves and salmon and cook until fish is opaque on one side, 1 to 2 minutes. Turn salmon, pour reserved dressing over it and cook until fillets are opaque on the surface and slightly translucent at the center, about 1 minute.
5. Flake salmon into chunks with a fork and add to potato and green bean mixture; toss with 3 tablespoons of the parsley. Spoon onto plates and garnish with remaining parsley and the lemon slices if desired; serve.