Preheat oven to 400°F and line baking sheet with parchment paper.
Cut delicata in half lengthwise; scoop out seeds and pulp. Cut lengthwise again, and then slice into 1/4-inch quarter moons. In a medium bowl, toss delicata with 1 tablespoon oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper until coated. Spread out in a single layer onto a baking sheet. Roast, stirring once or twice, until tender and darkened in spots, 16–20 minutes.
Meanwhile, combine vinegar, honey, caraway, the remaining 3/4 teaspoon salt, and the remaining 1/4 teaspoon pepper in a microwave-safe measuring cup or bowl. Microwave until boiling hot, 60–90 seconds. Remove from microwave and stir in shallots. Set aside for 10 minutes. Whisk in remaining 2 tablespoons of oil.
Toss cabbage, pomegranate, celery, hot roasted delicata squash, and warm shallot dressing in a large salad bowl. Divide among four plates and top with pecans and parsley.