Day 1. Rinse the beans, picking out any foreign matter; add enough water so that they are covered by about 3 inches, and soak overnight.
Day 2. Combine the first 5 ingredients to make the seasoning mix, and set aside.
Drain the beans and place them in a 4-quart pot with 1½ quarts of water, 2 tablespoons plus ¾ teaspoon of the seasoning mix, 1 cup of the onions, ½ cup red bell peppers, ½ cup yellow bell peppers, and ½ cup green bell peppers. Stir well, cover and bring to a boil over high heat. Reduce the heat to medium and cook, uncovering to stir and scrape the bottom of the pot occasionally, until the liquid is reduced so that it just covers the beans, and the beans are tender, about 1½ to 2 hours.
While the beans are cooking, start the dressing. Preheat the oven to 425°. Place the parsnips on a sheet pan, and brush all their surfaces with 1½ teaspoons of the oil. Roast in the oven, turning once or twice so they will brown evenly, until they are a medium golden brown, about 8 to 10 minutes. Set aside.
When the beans are cooked, transfer 1½ cups of them to a blender, and add the parsnips, the remaining 2 tablespoons of the olive oil, the rice vinegar, mirin, cane syrup, brown sugar, and the remaining 2 tablespoons plus 1¼ teaspoons of the seasoning mix. Process until smooth, about 30 to 40 seconds. Makes about 3¼ cups of dressing.
Place the remaining beans in a 2-quart mixing bowl and add the remaining, bell peppers, the celery, and the tomatoes. Toss to mix, then pour the dressing over the salad and toss gently until all the vegetables are coated with the dressing.