Step 1: Set a collapsible steamer basket in a large pot and add enough water to skim the bottom of the basket. Bring the water to a boil over high heat. Place the artichokes in a shallow layer in the basket. Steam, covered, adding more water as needed, until the leaves release easily when pulled and the hearts are tender when pierced with a paring knife, 15 to 20 minutes.
Step 2: Meanwhile, in a large bowl whisk together the lemon zest, lemon juice, balsamic vinegar, olive oil, garlic, fine sea salt, ⅛ teaspoon of pepper, and three quarters of the parsley.
Step 3: Add the steamed artichokes to the lemon marinade and toss to distribute the marinade evenly. Let stand until the marinade has infused the artichokes, at least 30 minutes or up to 3 hours. Alternatively, cover the bowl and refrigerate overnight.
Step 4: About 10 to 15 minutes before you plan to cook, heat a grill to medium-high heat.
Step 5: Use tongs to transfer the artichokes, cut side down, to the grill; keep the bowl of marinade next to the grill. Cook the artichokes until golden grill marks appear, 5 minutes, then flip them.
Step 6: Place the lemon quarters flesh side down on the grill. Cook the artichokes and lemons until they are lightly charred and the artichokes are completely tender, 3 to 5 minutes. Transfer the lemons to a serving platter. Return the artichokes to the bowl of marinade and toss to coat evenly. Arrange the artichokes on the serving platter, and sprinkle all with the remaining parsley, coarse or flaked sea salt, and additional pepper. Serve immediately.