Cut squid bodies along sides to open flat. With a thin, sharp knife, lightly score insides, being careful not to cut all the way through.
In a bowl, combine 2 tablespoons olive oil, lemon juice, garlic, and salt. Preheat grill to high heat.
Grill squid, turning occasionally, until opaque, slightly golden, and just cooked through, about 2 to 3 minutes. Remove squid to cutting board and slice into strips. Slice tentacles if desired.
Lightly drizzle with additional extra virgin olive oil as desired, sprinkle with and parsley, and serve with the lemon wedges on the side.