Mix the onion and vinegar in a small bowl. Cover and set aside for 10 minutes, stirring halfway through.
Combine the fennel, green beans, carrots, chickpeas, tomatoes, and radicchio in a large bowl. Gently toss.
Drain the onion slices, reserving the vinegar. Whisk the reserved vinegar with the olive oil. Add 1 ¼ tablespoons salt and ½ teaspoon pepper.
Drain the tuna and flake it into large chunks in a medium bowl. Toss with half the onion slices, 2 tablespoons of the vinaigrette, and the parsley.
Toss the remaining vinaigrette and onion with the chopped vegetables. Divide the vegetables among 4 serving bowls and top with the dressed tuna.