Place the scrubbed potatoes in a large pot and cover with cold salted water. Bring to a boil over high heat, then reduce to a simmer and cook for 20-25 minutes, until fork-tender but not falling apart. Drain and let cool just until you can handle them.

While potatoes cool, place the sliced red onion in a small bowl with cold water and a pinch of salt. Let soak for 10 minutes to mellow the bite, then drain and pat dry. Dice the tomatoes and cucumber, chop the parsley, and halve the olives.

In a small jar or bowl, whisk together the olive oil, lemon juice, red wine vinegar, salt, pepper, and pul biber (if using). Set aside.

Once potatoes are cool enough to handle, cut them into bite-sized cubes (about 1-inch). Place in a large mixing bowl. Add the drained red onion, diced tomatoes, cucumber, parsley, and olives. Pour the dressing over the top and toss gently to combine.

Let the salad sit at room temperature for at least 15 minutes to allow flavors to meld. Transfer to a white serving bowl, garnish with extra parsley and a sprinkle of pul biber, and serve at room temperature or slightly chilled.