Put the potatoes in a pot of generously salted water and bring to a boil. Cook until just tender, 8 to 10 minutes. Drain and cool slightly.
While the potatoes are cooking, whisk together the vinegar, mustard and garlic in a bowl, and season with salt and pepper. Slowly whisk in the oil until emulsified.
Heat the grill to high. Toss the potatoes with a little canola oil, and season with salt and pepper. Put the potatoes in a grill basket and grill, shaking the basket occasionally and turning the potatoes as needed, until charred in spots and just cooked through, about 5 minutes.
Put the potatoes in a serving bowl and toss with the dressing while still hot. Add the parsley and toss again. Serve.