In a large pot over medium-high, heat the oil until shimmering.
Add the garlic head cut side down and cook undisturbed until well browned, 2 to 4 minutes. Transfer the garlic to a small plate, then add the leeks, bay and ½ teaspoon salt to the pot. Cook, stirring occasionally, until the leeks are lightly browned, about 6 minutes.
Add the potatoes and cook, stirring occasionally, until lightly browned, about 5 minutes.
Add the water, cover and bring to a boil. Reduce to low, then nestle the carrots around the potatoes. Cover and cook until the potatoes are tender, 15 to 20 minutes.
Squeeze the root end of the garlic head to release the cloves into a small bowl; discard the head. Using a fork, mash the garlic into a paste. Stir the garlic paste into the soup and simmer, uncovered, over low for 5 minutes. Season with salt. Serve drizzled with additional oil and sprinkled with parsley.
