Combine basil leaves, parsley, olive oil, garlic, lime juice, salt, and pepper in a mini food processor and puree.
Set aside 2 tablespoons of the basil mixture.
Add shrimp to a large bowl.
Add remaining basil mixture to the shrimp and gently stir to combine. Cover and let sit 15 to 20 minutes. If using bamboo skewers, let them sit in water for at least 15 minutes before adding on the shrimp.
Preheat a grill to Medium-High.
Thread the shrimp onto the skewers.
Brush the grill grates with vegetable oil.
Grill the shrimp skewers for about 2 to 3 minutes per side, or until just charred and cooked through. When properly cooked, the shrimp exterior should be pink with red tails and the flesh will be slightly opaque in color. Do not cook for too long or the shrimp will be rubbery.
Remove shrimp from grill and brush with reserved basil mixture. Serve with lime wedges.