In a medium bowl, whisk together oil, lemon juice, and parsley until well combined; set vinaigrette aside.
Heat a grill that has a fine-mesh grill basket, or a grill pan, over medium-high heat. Place calamari, shrimp, and scallops in a medium bowl. Add 2 tablespoons reserved vinaigrette, and toss to coat. Grill seafood in batches until browned and cooked through, about 2 minutes on each side for shrimp and calamari, and 1 minute for scallops. Slice grilled calamari bodies into rings, and place in a clean bowl with remaining grilled seafood. Add shredded radicchio and arugula, and toss with remaining vinaigrette. Season with salt and pepper, and serve at room temperature on small plates or scallop shells.