1. In a large saucepan, combine the water, kosher salt, sugar, herbes de Provence, lemon zest and juice, smashed olives and bay leaves and simmer to dissolve the salt. Pour the brine into a very large, nonreactive bowl or stock pot and let cool to room temperature, stirring occasionally. Submerge the chicken in the brine and cover, or put the chicken and brine in a heavy-duty plastic bag. Refrigerate overnight.
2. In a small saucepan, whisk together the vegetable oil, pimentón de la Vera and 1 tablespoon of the chopped thyme. Place over moderate heat until hot.
3. Light a grill. Brush the onion slices with some of the paprika oil. Season with salt and pepper and sprinkle with the remaining 1 tablespoon of thyme. Grill the onions over moderately high heat, turning once, until lightly charred and crisp-tender, about 4 minutes per side. Transfer the onions to a large platter. Toss to separate the rings and drizzle with 2 tablespoons of the paprika oil. Add the sliced olives and parsley and toss again.
4. Remove the chicken pieces from the brine and pat dry. Season with salt and pepper. Grill the chicken over moderate heat, turning often, until the skin is nicely charred and the meat is cooked through, about 25 minutes. Toward the end of cooking, brush the chicken all over with the paprika oil. Transfer the chicken to the platter with the onions and serve.
MAKE AHEAD The smoked-paprika oil can be refrigerated for up to 1 week.