Heat oven to 450 degrees. Cut acorn squash in half lengthwise, scoop out seeds, and place, cut side down, on a baking sheet coated with olive-oil cooking spray. Roast until tender, about 45 minutes.
Combine chicken stock, apple cider, 1 teaspoon horseradish, salt, and pepper in a medium saucepan, and bring to a simmer. Scoop squash flesh out of skins, and place in the bowl of a food processor. Add 1 cup hot stock mixture, and puree until smooth. Stir squash mixture into pan with remaining stock, and keep soup warm.
Peel and core apples, and cut into 1/4-inch dice. Transfer to a small bowl, add lemon juice and remaining 2 teaspoons horseradish, and toss to combine. Heat a medium sautepan coated with olive-oil cooking spray over medium-high heat. Add apple mixture, and saute until golden brown. Serve soup garnished with sauteed apple mixture and chopped parsley.