Toss parsnips with 2 tablespoons olive oil in a bowl; spread parsnips onto a baking sheet.
Bake in the preheated oven until slightly tender, 20 to 25 minutes. Reduce oven heat to 350 degrees F (175 degrees C).
Toss apples and pears with 2 tablespoons olive oil in a bowl. Push parsnips to one side of the baking sheet and spread apples and pears onto the other side of the baking sheet.
Bake in the oven until fruit is soft and caramelized, 15 to 20 minutes.
Heat remaining olive oil in a large pot over medium heat. Cook and stir leeks in hot oil until leeks are soft, about 5 minutes. Sprinkle balsamic vinegar over leeks. Stir parsnips, apples, pears, chicken stock, and rosemary into leeks. Bring to a simmer and cook until fruits and vegetables are completely tender and easily mashed against the side of the pot, about 20 minutes.
Puree vegetable mixture using an immersion blender (or blend in batches in a blender) until soup has an applesauce-like consistency.