Mix butter, garlic, and parsley in a bowl. Set aside.
Heat oil in a skillet over medium-high heat. Season shrimp with paprika, salt, and pepper. Cook for 2-3 mins per side until pink. Remove from heat.
Butter one side of each bread slice with the garlic mixture. Sprinkle Parmesan on the buttered side.
On the unbuttered side, layer mozzarella, cheddar, shrimp, and more cheese.
Top with another slice of bread, buttered side facing out.
Grill sandwiches in a hot skillet for 3-4 mins per side until golden and cheese is melted. Cover skillet for even melting.
Cool slightly before slicing. Serve hot with marinara sauce or a fresh salad.