Prepare a two-zone fire for medium heat (350 degrees to 450 degrees F).
Cut the lemon in half and squeeze the juice into a large, nonreactive bowl; fill the bowl two-thirds full with water. Reserve the lemon halves. Working with one artichoke at a time, trim the very end of the stem (if attached) and remove the smallest leaves. Using scissors, snip off the thorny tips from the outer leaves. As you work, rub the cut surfaces with the pulp side of the lemon halves. Put the trimmed artichokes in the lemon water.
Cut off the tops of the garlic heads to expose the cloves; discard the tops. Place the garlic heads side by side on a square of aluminum foil. Wrap the foil around the garlic, like a canoe, leaving the tops exposed. Drizzle with 1 teaspoon of the oil and 1 teaspoon water and season with a pinch of salt and pepper.
Drain the artichokes. Place each artichoke on a square of aluminum foil. Drizzle the remaining 2 tablespoons of oil over the artichokes and season them evenly with 11/2 teaspoons salt and teaspoon pepper. Wrap each artichoke in its foil.
Brush the cooking grate clean. Drain and add one handful of the wood chips to the charcoal and put the lid on the grill. When the wood begins to smoke, cook the garlic and artichokes over INDIRECT MEDIUM HEAT, with the lid closed as much as possible, for about 40 minutes. Remove the garlic from the grill and let cool slightly.
Open the foil to expose the artichokes. Drain and add the second handful of wood chips to the charcoal. Continue grilling the artichokes until a large leaf can easily be pulled off, about 7 minutes more, turning the artichokes over once in the foil. Remove the artichokes from the grill.
Squeeze out the garlic cloves into a medium bowl. Mash the garlic with a fork. Add the mayonnaise and parsley and mix. Season with salt and pepper. Serve the artichokes warm or at room temperature with the aioli.