Preheat the oven to 425°F. Line a rimmed baking sheet with parchment paper.
Place potatoes in a large pot and cover with water. Add 1 tablespoon salt and bring to a boil over high heat. Reduce heat to medium-high; cook for 15 minutes, or until the potatoes are fork tender; drain.
Add potatoes in a single layer onto the pan. Using a heavy bottomed jar or glass, smash potatoes to ¼-inch thickness. Drizzle smashed potatoes evenly with olive oil and season with paprika, ½ teaspoon salt, and black pepper. Roast potatoes for 35 to 40 minutes, or until very crispy and browned.
While potatoes are roasting, prepare the dressing. Combine Greek yogurt or mayonnaise, olive oil, mustard, lemon juice, vinegar, garlic, chives, dill, and parsley. Stir well to combine and taste for seasoning and adjust as desired.
Reserve ¼ cup crispy potatoes for topping and set aside.
In a large bowl, combine remaining potatoes, carrots, green beans, and red onions. Stir in the dressing and toss to coat.
Add to a serving dish and top with reserved crispy potatoes and fresh parsley. Serve immediately. Enjoy!