Place potatoes in a large pot. Add enough cold water to cover and bring to a boil. Add 1 teaspoon salt, reduce heat and simmer until tender, 15 to 18 minutes.
Drain and run under cold water to cool. Cut potatoes in half (or quarter if large).
Meanwhile, in a large bowl, whisk together oil, vinegar, horseradish, 1/4 teaspoon salt, and 1/2 teaspoon pepper. Add potatoes to bowl and toss to coat. Fold in scallions, dill, and parsley (if using).
Make-Ahead Tip: Prepare without the scallions, dill, and parsley and refrigerate for up to 1 day. Fold in the scallions and herbs before serving