Slice Jerusalem artichokes into 1/4" slices, using a very sharp knife or mandoline.
Heat the olive oil and butter in a large skillet over medium-high heat. When hot, add the sliced artichokes, garlic, salt, pepper, and parsley. Stir to coat everything in oil.
Fry artichokes, stirring often, for about 4 minutes. Be careful not to overcook them - they should be slightly crunchy. Serve hot, as is, or with sour cream.