Combine potatoes and salt in a large pot filled with cold water. Bring to a boil. Cook until potatoes are tender, but not mushy, about 10 minutes depending on size. Drain and set aside until cool enough to handle.
While potatoes are cooking, whisk 3 tablespoons olive oil and white wine vinegar together in a small bowl. Add garlic and set aside.
Peel cooked and cooled potatoes and cut into 1-inch cubes. Combine potatoes, green onions, and parsley in a bowl and lightly toss. Season with salt and pepper. Drizzle with olive oil mixture and lightly toss, taking care not to break the potatoes.
Top with remaining 1 tablespoon olive oil and red pepper flakes right before serving.