Put the potatoes in a large pot and add enough water to cover by 1 inch. Season with salt. Bring to a boil over high heat and cook until the potatoes are fork-tender, about 10 minutes. Remove the potatoes with a slotted spoon and place them in a large bowl. Add the snap peas to the boiling water and cook until just crisp-tender, about 1 minute. Drain, then thinly slice on an angle. Add to the bowl with the potatoes; add the chopped celery. Meanwhile, make the dressing: Combine the mayonnaise, parsley, buttermilk, chives, lemon juice, tarragon, anchovies, shallots, avocado, celery leaves and 1 teaspoon salt in a blender and blend until smooth. Once the potatoes have cooled slightly, pour the dressing over the vegetables, season with the remaining 1⁄2 teaspoon salt and a few grinds of pepper and toss well. Cover and refrigerate until chilled, at least 1 hour or overnight. Toss the potato salad again and top with more parsley and chives before serving.