Heat the grill to high. Place the potatoes in a large saucepan and cover by 1-inch with cold water. Add 1 tablespoon of salt, bring to boil over high heat and cook until just cooked through, about 10 minutes. Drain, let cool slightly and halve.
While the potatoes are cooking, combine the mayonnaise, mustard, peppers, garlic, vinegar, paprika and salt and pepper in a blender or food processor and process until smooth.
Brush the cut sides of the potatoes with oil and season with salt and pepper. Grill cut side down until lightly golden brown, about 2 minutes.
Toss the potatoes with the red pepper mayonnaise while still warm, mashing slightly to break them up. Fold in the parsley and season with salt and pepper, if needed.