Boil the potatoes. Place potatoes in a large pot and add water to cover the potatoes. Season water with salt and bring to a boil. Reduce heat to medium-low and cook until potatoes are tender, about 12 to 15 minutes.
Cool. Transfer the cooked potatoes to a colander and drain and rinse them under cold water. Let cool for 5 minutes.
Smash! Using the smooth side of a meat tenderizer, smash each potato to about 1/2-inch thickness.
Heat the cooking oil. Heat olive oil in a griddle pan set over medium heat. Add thyme and continue to cook for 30 seconds to infuse the oil.
Fry the potatoes. Working in batches, cook the smashed potatoes, turning once until they're browned and crispy, about 3 minutes per side. Add more cooking oil if needed.
Season and serve. Transfer the fried potatoes to a serving plate and season with salt and pepper. Immediately sprinkle with parmesan cheese and allow it to melt. Add chopped fresh parsley and serve.