Drain the beans, then with cold water in a large pot and add the onion, carrot, garlic and bay leaf and cook until the beans are soft, about 1 to 1 1/2 hours. Drain the beans and pick out the onion, carrot, garlic and bay leaf. Place the beans in a bowl and toss with a couple tablespoons each of olive oil and balsamic vinegar. Fold in the red onion and thyme, and season with salt and pepper, to taste.
Preheat grill to high. Brush tuna and radicchio with oil and season with salt and pepper. Grill the tuna for 2 to 3 minutes per side, or until slightly charred and cooked to medium-rare doneness. Grill the radicchio, cut-side down, until slightly charred and wilted. Remove the tuna and radicchio from the grill. Dice tuna into bite-size pieces, and drizzle with balsamic vinegar. Slice radicchio into 1/4-inch thick slices. Fold tuna and radicchio into the white beans. Transfer the beans to a platter and sprinkle with chopped parsley. Serve warm or at room temperature.