Put saffron threads in a small bowl, then stir in lukewarm water and let stand. Trim cauliflower stalk flush with base of crown, discarding stalk, then quarter cauliflower lengthwise. Cut each quarter lengthwise into 1/2-inch-thick slices. Set aside large slices and reserve small pieces without any stem for another use. Swirl oil with garlic in a deep 12-inch heavy skillet to coat, then heat over moderately high heat until oil is just smoking. Discard garlic. Carefully arrange cauliflower slices in 1 layer in skillet and cook until undersides are lightly browned, about 2 minutes. Turn over with tongs and cook until other sides are lightly browned, about 2 minutes. Add carrot and wine and swirl skillet to combine, then boil until liquid is reduced by half, about 30 seconds. Add saffron water, broth, 3/4 teaspoon salt, and 1/4 teaspoon white pepper and simmer, covered, until cauliflower and carrot are crisp-tender, 5 to 8 minutes. While cauliflower simmers, pat fish dry and sprinkle both sides with remaining 1/2 teaspoon salt and 1/4 teaspoon white pepper. Increase heat to moderately high, then add mussels to skillet and steam, covered, until mussels just open wide, checking frequently after 3 to 4 minutes. (Discard any mussels that remain unopened after 8 minutes.) Divide mussels among 6 shallow serving bowls using a slotted spoon. Place cod on top of cauliflower, arranging fillets in 1 layer. Sprinkle with scallions and parsley and steam, covered, over moderately high heat until fish is cooked through, 7 to 10 minutes. Transfer some of cauliflower along with 1 piece of cod to each bowl using slotted spoon. Season broth with salt and white pepper, then ladle broth (with carrots) over each serving.