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Preheat the oven to 275°F. Place the salmon fillets in a baking dish, skin side down, so the fillets are not touching. Season the fillets with salt and pepper, pour 2 tablespoons of the oil over the fish, add 2 tablespoons of water to the dish, and bake until the fillets are still bright in color and appear medium-rare and slightly undercooked when gently prodded, about 15 minutes. Remove from the oven.
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Meanwhile, bring a pot of salted water to a boil, add the green beans, and cook until crisp-tender, 3 to 4 minutes. Drain.
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In a small bowl, mix together the remaining 2 tablespoons oil, the vinegar, shallot, tomatoes, and 1⁄2 teaspoon salt until well combined. In a small skillet, simmer the tomato mixture and the green beans for 2 minutes. Stir in the parsley and remove from the heat.
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Place each fillet on a dinner plate. Spoon half of the tomato mixture over each fillet and serve