Preheat the oven to 200 degrees F.
Add the potatoes to a saucepan and cover with cold water. Bring up to a boil, reduce to a simmer and cook until tender, about 20 minutes.
Drain the potatoes in a colander and allow to cool slightly. Transfer to a large bowl and mash with a potato masher until smooth. Add the Parmesan, sour cream, mustard, green onions and egg. Sprinkle with salt and pepper. Mix together until incorporated.
Combine the panko and dried parsley in a casserole dish and sprinkle with salt and pepper. Form the potatoes into 8 cakes, about 5 inches wide and 3/4-inch thick. Dredge the cakes through the panko.
Add the canola oil to a high-sided skillet over medium heat. Once the oil is hot, add the first batch of cakes and cook until crisp, about 3 minutes per side. Remove from the oil to a paper towel-lined plate. Repeat with the second batch. Hold in the oven before serving.
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