Whisk together mayonnaise through mustard in a small bowl or process in a small food processor; set aside.
Off heat, coat a grill or grill pan with cooking spray; heat to medium-high heat.
Remove any damaged outer leaves from endive and radicchio; cut each in half lengthwise through core. Coat cut surfaces of endive and radicchio with cooking spray; season with salt and pepper.
Place capers in a small baking pan; coat with cooking spray. Place pan directly on grill; cook capers, stirring occasionally, until capers are lightly browned and crispy, 5-10 minutes.
Add endive and radicchio to grill, cut sides down; cook until lightly charred, 2-3 minutes. Flip and grill until slightly softened, 1-2 minutes. Remove from grill; brush cut sides with balsamic vinegar.
To serve, place 1 endive half and 1 radicchio half on a serving plate; drizzle with 1 Tbsp dressing, 1 Tbsp Parmesan, 1 Tbsp capers and 1/2 Tbsp parsley.