Soak the chickpeas in 1 quart cool water in a large container (they will more than double in size) overnight in the refrigerator. Drain. Bring the chickpeas, baking soda, and 1 fresh quart of water to a boil in a large lidded pan over high heat. Reduce the heat and simmer with the lid ajar, stirring occasionally, until the beans are tender. It should take about 1 hour.
Meanwhile, heat the oven to 400℉. Toss the carrots on a sheet pan with 1 tablespoon oil and ½ teaspoon salt. Roast for 25-30 minutes, turning once, until very tender and just a tad browned. Let cool.
Drain the chickpeas, reserving ½ cup of the cooking liquid, and cool. Set aside a few chickpeas for garnish.
Process the chickpeas, garlic, ½ teaspoon salt, cumin, and cayenne in a food processor or blender until uniformly ground. Scrape down the bowl. With the machine running, add the following ingredients thru the feed tube one at a time, stopping to scrape the bowl as needed: ¼ cup lemon juice, ¼ cup reserved cooking water, the cooled carrots, 3 Piquillo peppers, 2 tablespoons olive oil and tahini. Add more liquid if needed, until the hummus is smooth and fluffy. Adjust the olive oil, Piquillos, salt, lemon & cayenne to your taste.
Cover and let the flavors mingle for at least 30 minutes. Taste and adjust again. Drizzle with olive oil and sprinkle with the reserved chickpeas and parsley before serving.
Store for up to 5 days in the refrigerator.