Chop the celery into roughly the same-sized pieces, about an inch - 1.5 inches, Don't worry about precision as the soup is going to be puréed anyway. You just want it to cook evenly.
In a large soup pot, heat the oil over medium heat. Add the onion and cook until soft.
Add the garlic and cook another minute before adding the celery, stock, potato, and parsley. Increase the heat to medium-high and bring to a boil.
Cover, lower the heat and simmer for about 30 minutes or until the celery and potatoes are soft enough that they can easily be pureed (But don't let them get too mushy).
Remove the soup from heat. If necessary, using a ladle, remove enough stock so that the celery is just covered. Set any reserved stock aside.
Remove some of the chopped celery with a slotted spoon (1-2 spoonfuls) and set aside.
Purée the soup with a stick blender in the pot - adding the reserved stock little by little until the soup reaches the desired consistency.
Add the reserved celery, mix to combine and serve hot.
If you don't have a stick blender, a food processor or blender would work just as well to puree the soup.
To avoid stringy celery, use the more tender celery stalks that are closer to the middle of the head. Lighter colored celery is less likely to have though strings.
To test if your celery is stringy (which for some can be unpleasant), before cutting the celery, break it in half by hand. If you see the strings sticking out of the break (see photo above), peel the celery to remove the strings.
To remove the strings from the celery, clean and peel the back of the stalk with a vegetable peeler (for more detailed instructions, you can see this video on YouTube) or just by snapping your celery into sections and pulling out the exposed, stringy ribs.