Dice the potatoes and chop the leeks, garlic, and parsley.
Heat the oil in a soup pot over medium-high heat. Add the leeks and cook for 5 minutes. Add the garlic and cook 1 more minute.
Add the potatoes, broth, salt, and pepper. Bring the soup to a boil, then reduce heat to low and cook for 20 minutes or until the potatoes are fork-tender.
Add the spinach and parsley to the pot, stir, and cook for 2 minutes. Add the lemon juice to the pot.
Remove the pot from the stove and blend the soup with an immersion or upright blender until smooth.