Preheat the oven to 400 degrees F. Toss both kinds of potatoes, the parsnips, onion, garlic, olive oil, 1 teaspoon salt and 1/4teaspoon pepper in a bowl. Spread in a single layer on a baking sheet; bake until tender, about 45 minutes. Let cool slightly.
Meanwhile, mix 2 tablespoons water and the sugar in a skillet and bring to a boil over medium heat. Cook, swirling the pan, until amber, about 7 minutes. Remove from the heat and stir in the vinegar and 1/4teaspoon each salt and pepper. Pour the caramel into a 9-by-13-inch glass baking dish and spread with a rubber spatula. Sprinkle the sage and oregano on top.
Arrange the roasted potatoes and parsnips in a single snug layer on top of the caramel. Scatter the onion and garlic over the roasted vegetables; sprinkle evenly with the mozzarella.
Roll out the puff pastry on a lightly floured surface into a 9-by-13-inch rectangle. Pierce the pastry all over with a fork, then lay it on top of the mozzarella, folding the edges under to fit, if necessary. Bake 20 minutes, then reduce the oven temperature to 350 degrees F and continue baking until the dough is cooked through, 15 to 20 more minutes.
Let the tart cool 10 minutes in the baking dish, then carefully invert it onto a cutting board. Replace any vegetables that stick to the dish, if necessary.