3 large cedar planks for grilling, a grill basket and an instant-read thermometer
Soak 3 large cedar planks for grilling in water for 30 minutes. Prepare a grill for indirect cooking for the salmon and direct cooking for the tomatoes. Preheat a grill basket.
Sprinkle both sides of the salmon fillets with salt and pepper, then place them on the soaked planks skin-side down. Sprinkle the top of each with 1 teaspoon of sugar. Grill the salmon on the planks over indirect heat until just cooked through and the internal temperature registers 135 degrees F with an instant-read thermometer, 25 to 30 minutes.
Meanwhile, toss the cherry tomatoes with the olive oil and some salt.
When the salmon fillets are ready, leave them on the planks and transfer the whole thing to a serving platter; loosely tent with foil.
Add the cherry tomatoes to the grill basket and cook, stirring occasionally, until they are just blistered and some are starting to burst, 2 to 3 minutes. Carefully remove the tomatoes to a mixing bowl and toss with the parsley, lemon juice, 1/2 teaspoon salt and a few grinds of pepper.
Spoon the tomato mixture over the salmon and serve immediately.