Place potatoes in a large pot; cover with cold salted water. Bring to a boil, reduce heat, and simmer until tender, 15 to 18 minutes. Drain.
Arrange oven rack approximately 6 inches from top of oven and preheat to 450°F. Rub 2 tablespoons oil on 2 large rimmed baking sheets, dividing evenly. Divide potatoes between baking sheets. Working one at a time, gently press potatoes, with the bottom of a glass or your palm, until crushed but still intact. Brush potatoes with remaining 2 tablespoons oil, dividing evenly. Season with salt and pepper.
Roast, one pan at a time, turning potatoes once, until golden brown and crisp, 22 to 25 minutes.
Meanwhile, whisk together buttermilk, mayonnaise, honey, and thyme in a bowl. Season with salt and pepper. Add warm potatoes and toss to coat. Fold in scallions, red pepper, and parsley.
