Preheat the grill to medium-high heat.
Using kitchen scissors, carefully cut the top shell of each lobster tail down the center, stopping just before the tail fin.
Gently pull the lobster meat out of the shell and lay it on top, keeping it attached at the base.
In a small bowl, melt the butter in the microwave or on the stove. Add minced garlic, lemon juice, paprika, salt, and pepper. Mix well.
Brush the butter mixture generously over the lobster meat.
Place the lobster tails meat-side up on the grill. Close the lid and grill for about 6-8 minutes, or until the meat is opaque and cooked through.
Carefully remove the lobster tails from the grill using tongs. Brush with any remaining butter mixture.
Transfer the lobster tails to a serving plate and garnish with chopped parsley.