Heat a grill to high heat. Rinse the squid under cold running water and pat dry with paper towels.
Cut the squid bodies lengthwise down one side and open flat.
Cut the tentacles in half if too large.
In a bowl, combine the olive oil, lemon juice, salt, and pepper.
Add the squid bodies and tentacles to the bowl, tossing to evenly coat.
Thread the squid bodies lengthwise onto skewers so they lie flat. Thread the tentacles onto separate skewers.
Grill over high heat, turning once, just until opaque throughout, about 1 to 2 minutes.
Remove the squid from skewers and pile them on a platter.
Sprinkle with parsley and serve with lemon wedges.