Pour the oil into a heavy, deep-sided skillet or deep fryer; place over medium-high heat until it registers 375 degrees on a deep-fry thermometer. Pour buttermilk into a medium bowl.
To prepare artichokes, snap off tough outer leaves. Slice off the upper two-thirds of each bulb, and discard. Use a vegetable peeler to remove green outer layer from around heart. Dip in buttermilk, and scoop out choke with a spoon. Thinly slice crosswise. Place in the buttermilk.
Add the fennel and lemons, and stir to coat evenly. Place flour in a separate dish. Season generously with salt and pepper; whisk to combine.
Remove vegetables and lemon slices, one piece at a time, from buttermilk, shaking off excess. Dredge in flour, and carefully transfer to the hot oil, 6 to 7 pieces at a time. Cook until golden brown, about 3 minutes. Dip parsley in buttermilk, and then in flour to coat. Carefully transfer to the hot oil, and fry until golden brown, about 30 seconds. As each batch is finished, transfer to a paper towel-lined baking sheet to drain. Sprinkle with salt. Serve immediately.