Special equipment: an 8-by-12 glass baking dish
Position 2 oven racks in the center and upper third of the oven, and preheat to 425 degrees F.
Toss the parsnips, onions, oil and 1 teaspoon salt in a large bowl, spread them out in a single layer on a rimmed baking sheet, and bake on the upper rack, without moving the vegetables, until tender and browned on one side, about 30 minutes. Toss with the rosemary.
Meanwhile, toss the apple wedges, butter and 1/2 teaspoon salt in a medium bowl, and spread them out in a single layer on a second rimmed baking sheet and bake on the center rack, without moving the apples, until slightly softened and browned on the bottom, about 15 minutes. When the vegetables and apples are done, remove them from the oven and lower the heat to 400 degrees F.
While the vegetables and apples cook, sprinkle the sugar and a pinch of salt on the bottom of a large skillet. Cook, undisturbed, over medium heat until the sugar melts, 6 to 8 minutes, then swirl until it caramelizes and turns golden amber, about 4 minutes more. Remove from the heat, and carefully whisk in the vinegar to avoid spattering. If the caramel hardens, return it briefly to the heat, and stir to melt. Pour the caramel into an 8-by-12-inch glass baking dish, and spread with a heat-resistant spatula to coat the bottom.
Arrange the parsnips and apples (alternating) in the caramel, browned-side down. Nestle the onions into the gaps, and cover with an even layer of Parmesan.
Lightly dust a work surface with flour, then roll out the puff pastry to 8 by 12 inches. Poke the dough all over with a fork, then lay it on top of the Parmesan. If the dough doesn't quite fit, fold any overhang under the dough edges or stretch the dough slightly to the edges.
Bake the tart on the middle rack until the dough just starts to puff, 8 to 10 minutes; reduce the heat to 350 degrees F, and continue to bake until the dough is cooked through and golden brown, 20 to 25 minutes more, rotating the dish halfway through. Let cool for 5 minutes, then quickly but carefully (to avoid being burned by the dripping hot sugar) flip the baking dish well away from you onto a flat serving platter or cutting board. Serve warm or at room temperature.
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