Preheat the oven to 350°F. Spread the carrots and parsnips onto a baking sheet in one even layer. Drizzle with olive oil and season with salt and pepper. Roast for about 20 minutes, until the vegetables are tender and just beginning to caramelize. (Give the vegetables a toss 10 minutes into the cooking process.)
Meanwhile, in a small saucepan, bring the apple juice, apple cider vinegar, honey, and molasses to a boil over medium-high heat. Reduce to a simmer and let the mixture cook down until it has thickened enough to coat the carrots and parsnips as a glaze.
In a food processor, whip the goat cheese while drizzling in the olive oil. Add the salt and pepper. Process until smooth.
Toss the roasted carrots and parsnips with ¾ of the sweet apple glaze. Check the vegetables for seasoning, adding more salt or pepper if necessary. Spread the whipped goat cheese onto a plate and place the carrots and parsnips on top. Spoon the rest of the glaze over the entire dish.