Preheat the oven to 400 degrees. Cut the potatoes into one inch cubes. Place the potatoes in a large bowl and toss with the vegetable oil until well coated.
Cover one or two baking sheets with foil and coat lightly with non-stick spray. Spread the potatoes out over the sheets so that they are in a single layer and not piled on top of one another (you may need two baking sheets if your sheets are smaller). Season lightly with salt and pepper, then roast in the oven for 45 minutes or until golden brown and crispy on the edges.
Meanwhile, rinse the parsley with cool water and shake off as much excess water as possible. Tear the leaves from the stems and place them in the bowl of a food processor (it's okay if some of the stems get in, you don't have to be very precise). Also add the Parmesan, salt, lemon juice, and peeled garlic. Pulse the mixture until it is finely chopped.
Scrape down the sides of the bowl and slowly add the olive oil through the chute while pulsing. Scrape down the sides again and add one tablespoon of water while pulsing. The pesto should be a thick, almost creamy texture at this point. Taste the pesto and adjust the oil, salt, and lemon according to your liking (depending on the size of your parsley bunch, you may need to adjust the other ingredients).
Once the potatoes are golden brown and crispy, remove them from the oven and transfer them to a bowl. Serve the pesto spooned over top or toss the potatoes in the pesto until they are fully coated.