Heat the oven to 400°F. On a small rimmed baking sheet, toss the potatoes with 1 Tbs. of the oil and season generously with salt and pepper. Roast until tender when pierced with a skewer, 25 to 30 minutes. Cool the potatoes; they can be served still warm or at room temperature.
Meanwhile, bring a medium saucepan of salted water to a boil. Cook the haricots verts until just tender (taste one to see), 3 to 4 minutes. Drain in a colander and run under cold water until cool.
Combine the yogurt and parsley and season well with salt and pepper; keep chilled until ready to serve.
To serve, halve or quarter any larger fingerlings but leave the tiny ones whole. Combine the potatoes, haricots verts, celery, cucumber, fennel, and red onion in a large bowl and toss with the remaining 3 Tbs. oil and the lemon juice. Season well with salt and pepper and toss again. Taste and add more lemon juice or olive oil if needed.
Divide the salad among six to eight plates, piling it toward the center of each plate. Drop a large dollop of yogurt next to each salad.
You can prepare both the yogurt and the vegetables a few hours ahead of serving. Stir together the yogurt, parsley, and seasonings and refrigerate, covered. Combine all the vegetables but the potatoes in a bowl but don’t dress them. Cover the vegetables with a damp paper towel and keep refrigerated; leave the potatoes at room temperature.