Heat oven to 450 degrees.
Roast the vegetables:
Place beets on parchment-lined aluminum foil and drizzle with 1 tablespoon olive oil. Wrap in foil and bake until beets are tender when pierced with the tip of a knife, 30 to 45 minutes for baby beets and up to 1 3/4 hours for larger beets. Let stand until cool enough to handle; rub off skins with latex gloves or paper towels. Cut beets in half or into quarters or sixths if large. Transfer to a large bowl; set aside.
Toss peeled shallots, parsnips, carrots, and rosemary in a large bowl with remaining 3 tablespoons oil, then season with salt and pepper. Spread in a single layer on a rimmed baking sheet. Roast until tender and golden, turning vegetables over once, about 30 minutes. Transfer to a large bowl; set aside.
Make the cider vinaigrette:
Combine cider, mustard, shallot, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a bowl and whisk to combine. Whisking constantly, add olive oil in a slow, steady stream until oil is incorporated and mixture is emulsified.
Wash arugula and dry thoroughly, then place in a bowl and cover with a damp kitchen towel or damp paper towels. Refrigerate until needed. This will help crisp the leaves.
Toss parsnips, carrots, and shallots with 2 tablespoons vinaigrette. Toss beets separately with 1 tablespoon vinaigrette to prevent their color from bleeding. Toss arugula with 2 tablespoons vinaigrette.
Line a serving platter with arugula and arrange vegetables on top. Serve immediately.