Bring a large pot of salted water to a boil. Add cauliflower; cook until crisp-tender, about 4 minutes. Using a slotted spoon, transfer cauliflower to a large bowl.
Add pasta to same pot; cook until just tender but still firm to the bite. Drain, reserving 1 cup pasta cooking liquid.
Meanwhile, heat 3 tablespoons oil in a large skillet over medium heat. Add garlic; stir until fragrant, about 1 minute. Add chickpeas; cook, stirring occasionally, until beginning to brown, about 3 minutes. Add 1 cup pasta cooking liquid and cauliflower. Simmer until liquid is slightly reduced, about 4 minutes. Remove from heat; stir in arugula, lemon juice, 2 tablespoons butter, and remaining 3 tablespoons oil. Season to taste with salt and pepper.
Add pasta to skillet with cauliflower mixture and toss to coat, adding more pasta cooking liquid by ¼-cupfuls as needed. Divide pasta among bowls. Garnish with capers and remaining 2 tablespoons butter. Serve topped with Parmesan.
