PREPARE THE DRESSING: (Half will be used to grill shrimp. Half will be used for making the shrimp salad.) Mix all dressing ingredients in a medium bowl. Set aside 2 1/2 tablespoons in a second medium bowl to dress the grilled shrimp.
PREPARE AND GRILL SHRIMP: Heat grill to high. Mix shrimp with 2 1/2 tbsp of dressing. Thread onto steel skewers or wooden skewers that have been soaked in water for a few minutes. Spray shrimp with oil to prevent sticking. Grill shrimp for about 2 minutes per side or just until cooked through. Smaller ones will take less time. Do NOT overcook.
MAKE THE SHRIMP SALAD: Remove shrimp from skewers to cutting board, Roughly chop into chunks and add to bowl with reserved dressing. Add celery and parsley. Mix to combine. Taste and adjust seasonings (I add another good squirt of lemon juice).
TOAST ROLLS AND STUFF THEM: Heat a large pan to medium. If rolls have crust on the sides, cut a thin slice off. Generously butter sides of rolls. Place in pan and toast for a couple minutes on each side until golden brown. Add about 3/4 cup shrimp salad to each roll. Serve with pickles and chips or salad if desired.
Make Ahead: The shrimp salad can be made ahead and refrigerated for several hours or even a day.