Set a large bowl or stockpot half full of water. You may need to use two pots if your artichokes are large. Add the juice of a half of a lemon into the stockpot. Clean the artichokes and remove the outside, lower leaves. Trim the stem, but leave a few inches for grilling as they will stay together better. Use a very sharp knife to trim the prickly edge off the tips of each artichoke.
Place them face down into the lemon water and soak for 5 or 10 minutes. Pour out the water and replace with fresh water. Remove one artichoke at a time from the water to a small plate. Stuff the artichoke with at least one large clove minced or crushed garlic, trying to push some into the inside, and some between the leaves. This works best with a dinner knife, to push it down. Then set gently back into the water, face up. Continue for remaining artichokes.
Fill the stockpot with additional water to cover the artichokes. The artichokes will pop up out of the water. Add 1/2 -1 teaspoon of salt into the stockpot. Add the parsley sprigs to the water. Finish by drizzling 1/2 - 3/4 teaspoon olive oil in the center of each, while trying to keep them standing upright.
Cover with the lid cracked and set on the burner and bring to a simmer. Simmer for 1 1/4 - 1 1/2 hours or until the outside artichoke leaves are tender to serve whole or 45 minutes to 1 hour to cut into halves and finish on the grill. **See note below To Grill Halves of Artichokes after simmering. Reduce the cooking time to 35-40 minutes.
