1. Preheat an outdoor grill to medium-high heat. Whisk together 1/4 cup of the olive oil, anchovies, and orange juice in a bowl to make the dressing. Set aside.
2. Toss the radicchio quarters in a bowl with the remaining 1 1/2 tablespoons oil. Grill until lightly charred and beginning to wilt, turning occasionally, 7 to 10 minutes total. Meanwhile, brush or drizzle the bread with olive oil and toast alongside the radicchio.
3. Remove the radicchio and chop into bite-size pieces, discarding the core. Toss with the dressing and sprinkle with the parsley. Divide among 4 plates and shave the Parmesan over the top with a vegetable peeler. Serve with the grilled bread.
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