Slice radishes into very thin rounds. Using a vegetable peeler, peel carrots lengthwise into long, thin slices. Combine 2 tsp. salt and 3 cups ice water in a medium bowl and soak radishes and carrots for 10 minutes. Drain and pat dry. Meanwhile, whisk lemon juice, vinegar, mustard, pepper, and remaining 1/4 tsp. salt in a small bowl until combined. Whisk in oil in a slow stream. Toss radishes, carrots, and capers in a large bowl with just enough vinaigrette to coat. Divide among 4 plates and top with tuna and parsley; drizzle with remaining vinaigrette and top with a few grinds of black pepper.