In a large saucepan or Dutch oven combine sliced potatoes and salt. Cover with water by 1 inch. Bring to a boil over high heat, then reduce heat to medium-low and cook until potatoes are easily pierced, about 5 to 6 minutes.
Reserve ¼ cup cooking water, then drain. Transfer the potatoes to a large mixing bowl.
In a small food processor or blender, combine the olive oil, parsley, green onions, lemon juice, Dijon mustard, garlic, and freshly ground black pepper. Process until the herbs and garlic have been chopped into little pieces. Then, while running the food processor, pour in the reserved cooking water and process just until blended.
If you don’t have a food processor or blender, just finely chop the parsley and onions and whisk the dressing together until the oil is fully incorporated.
Drizzle the potatoes with the herbed olive oil mixture and gently mix to combine. (It will look like you’ve poured in too much dressing, but don’t worry, the potatoes will soak it up!) Let the potatoes rest for 10 minutes, gently tossing every few minutes. Add the celery to the bowl and give it a final toss.