Preheat oven to 450°F. Cut the fennel bulb in half, remove its core, and then cut each half into quarters and separate the layers.
Place fennel, carrots, parsnips, and beets on a rimmed baking sheet. Drizzle with 2 tablespoons oil and season with ¾ teaspoon salt and the pepper. Toss to coat and then spread in an even layer. Roast until tender when pierced with a fork and golden brown in spots, 25 to 30 minutes.
Meanwhile, stir garlic and ¼ teaspoon salt in a small bowl, and then mash with a fork until a paste forms. Add lemon zest and juice, tahini, and remaining 1 tablespoon oil, whisking until smooth. Stir in parsley, poppy seeds, and remaining ¼ teaspoon salt. Stir in 1 to 2 tablespoons warm water until dressing reaches desired pourable consistency.
Drizzle half of the dressing over roasted vegetables, and then serve remaining dressing alongside. Top with fennel fronds.
