Preheat the oven to 425°F. Line a large rimmed sheet pan with foil or parchment and lightly oil the surface so vegetables don’t stick.
Toss the diced potatoes with 1 tbsp of olive oil, 1/2 tsp salt and a few grinds of black pepper. Spread them in a single layer on one side of the sheet pan. Roast for 15 minutes to start crisping the edges.
While the potatoes roast, whisk together the Dijon, 1 tbsp olive oil, the zest and juice of the lemon, and the minced garlic in a small bowl. Taste and adjust with a pinch of salt if needed — this is the glaze for the salmon.
Toss the trimmed green beans with the remaining 1 tbsp olive oil and a light pinch of salt and pepper. After the potatoes have had 15 minutes, move them to one side of the pan to make room.
Nestle the green beans on the empty side of the pan. Pat the salmon fillets dry and place them skin-side down in the center space between the potatoes and beans. Brush the tops of the salmon generously with the lemon-Dijon glaze.
Return the sheet pan to the oven and roast everything for 10–12 minutes, or until the salmon reaches about 125–130°F and flakes easily. If you like crisp skin, switch the oven to broil for 1–2 minutes at the end — watch closely so it doesn't burn.
Remove the pan and let the salmon rest for 3–5 minutes. Sprinkle the whole pan with chopped parsley and a little extra lemon zest before serving. Divide salmon, potatoes and beans among plates and spoon any pan juices over the top.