Preheat a grill to high heat (450–500°F/230–260°C). Set a large cast-iron skillet or sturdy disposable rimmed pan on the grates and pour in the coarse kosher salt to make a level bed that will keep shells steady.
Melt the unsalted butter in a small saucepan over low heat until just liquid, 2–3 minutes. Add the garlic and warm gently until fragrant, 30–60 seconds; remove from heat and stir in the lemon (zested and juiced), flat-leaf parsley, hot sauce, kosher salt, and black pepper.
Shuck the oysters in shell, working over a bowl to catch any juices, and loosen each from the bottom shell while keeping them nestled in the deep half.
Set them into the salt-lined pan so they sit level. Spoon about 1–2 teaspoons of the warm mixture over each and sprinkle the parmesan cheese evenly on top.
Slide the pan to a hot spot on the grill, close the lid, and cook until the oysters are plump, edges just curl, and the tops are bubbling with light browning, 4–6 minutes.
Carefully transfer the pan from the grill. Let rest 1 minute, then serve immediately with lemon wedges (for serving).