Prepare the squid: Rinse well. Pull out and discard the quill and innards, remove the beak from the tentacles, peel the skin if desired, and separate tubes from tentacles; pat everything very dry.
Butterfly each tube by cutting it open lengthwise and lay it flat. Lightly score the inside surface in a shallow crosshatch (do not cut through).
In a large bowl, whisk the lemon juice, lemon zest, garlic, dried oregano, kosher salt, black pepper, and 3 tbsp of the olive oil; reserve the remaining 1 tbsp for finishing.
Add the prepared pieces to the bowl and toss to coat. Marinate at room temperature for 20–30 minutes.
Heat a grill to high (450–500°F). Scrub the grates clean, then oil them by rubbing with a folded paper towel dipped in the vegetable oil held with tongs.
Shake excess marinade from the pieces. Lay the tubes scored-side down and the tentacles directly on the grill, perpendicular to the grates.
Grill until the underside is well marked and the flesh turns opaque at the edges, 2–3 minutes. Flip and grill until just opaque throughout and lightly charred at the tips, 1–2 minutes more; do not overcook.
Transfer to a warm platter. Drizzle with the reserved 1 tbsp olive oil, sprinkle with parsley, and serve immediately with lemon wedges.