Heat oven to 400°F.
Toss together potatoes with the oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Spread out in an even layer on a large rimmed baking sheet.
Roast potatoes for 13 to 15 minutes, or until just tender and starting to brown a bit.
Remove from oven; push potatoes to one side of the sheet pan and arrange salmon fillets down the center.
Melt butter in a small mixing bowl and whisk in garlic, parsley, lemon juice, and brown sugar. Remove 1-1/2 tablespoons of the butter-garlic mix and set aside.
Rub salmon evenly with the remaining butter mixture.
Add the green beans to the other side of the pan and toss with the 1 1/2 tablespoons melted butter mix.
Season everything with the remaining salt and pepper.
Return to oven and continue baking for 10 minutes, or until the potatoes are golden and fork-tender and salmon is cooked through. Use an Instant Read Thermometer to check for doneness; salmon is done at 145˚F.
If you want extra browning on top of the salmon, put pan under the broiler for the last 2 to 3 minutes of cooking.
Serve with lemon slices.