Place potatoes in a large saucepan. Cover with cold water and add 1 teaspoon salt. Bring to a simmer over medium-high heat. Cook until fork-tender, about 10 minutes, adding green beans the last 2 minutes or cooking. Drain; transfer to a bowl of ice water. Drain well and set aside.
In a screw-top jar combine the oil, vinegar, mustard, and 1/4 teaspoon salt. Arrange potatoes, green beans, lox, lettuce, eggs, olives, and parsley on plates. Drizzle each with dressing.