Preheat your grill to medium-high heat (about 350°F to 400°F / 175°C to 200°C).
Using kitchen scissors, cut the top shell of each lobster tail down the center, stopping just before the tail. Gently pull apart the shell to expose the lobster meat.
In a small bowl, melt the butter and stir in the minced garlic, chopped parsley, lemon juice, paprika, salt, and pepper.
Brush the lobster meat with a generous amount of the garlic butter mixture, reserving some for basting during grilling.
Place the lobster tails on the grill, shell side down. Grill for about 5 minutes.
Flip the lobster tails and grill for another 5 minutes, basting with the remaining garlic butter. The lobster is done when the meat is opaque and cooked through.
Remove the lobster tails from the grill and let them rest for a couple of minutes. Serve hot on a platter with additional garlic butter drizzled on top if desired.