Put the cumin in a small skillet over low heat and toast, stirring frequently, until fragrant but not dark, 2 minutes. Remove to a bowl, add the olive oil, vinegar and salt and pepper and whisk to combine. Let sit at room temperature to allow the flavors to meld.
Put the potatoes in a pot, cover with cold water, add 1 tablespoon of salt, bring to a boil and cook until a paring knife inserted meets with some resistance, about 8 minutes. Drain and transfer to a bowl.
Heat a charcoal or gas grill to high.
Using a potato masher, smash the potatoes, leaving them very chunky, and toss with the canola oil and some salt and pepper. Put the potatoes in a grill basket and grill, covered, tossing occasionally, until slightly charred and crisp in spots, about 6 minutes. Remove to a bowl and toss with the vinaigrette. Taste, season as necessary and toss again with the chopped parsley.