Rinse and dry the potatoes. Trim off any bad parts (do not peel). Cut the potatoes in half lengthwise.
Transfer the potatoes to a large bowl. Add the oil, garlic, ½ teaspoon salt, thyme, and pepper. With your fingers (easiest) or a large spoon, toss to combine.
Transfer the potatoes to a large rimmed baking sheet or roasting pan. For best results, flip the potatoes so the cut sides are flat against the pan. Make sure the pieces do not touch.
Bake until the potatoes are browned and crisp and tender enough to easily pierce with a fork, 30 to 40 minutes depending upon the size of the potatoes. Flip the potatoes twice during cooking, spreading them back into an even layer each time.
Remove the potatoes from the oven. Sprinkle with the parsley, Parmesan (if using), and remaining 1/4 teaspoon kosher salt. Stir to combine. Serve hot.
TO STORE: Refrigerate leftover potatoes in an airtight storage container for up to 4 days.
TO REHEAT: Rewarm potatoes on a baking sheet in the oven at 350 degrees F or in the microwave.
TO FREEZE: Freeze leftovers in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.