Heat 1 1/2 inches oil to 375°F in a 2 1/2- to 3-quart saucepan, then fry parsley (use caution; oil will spatter) until leaves are dark green, about 45 seconds.
Transfer with a slotted spoon to several layers of paper towels and season with salt.
Return oil to 375°F, then fry lemon slices, stirring occasionally, until golden in spots, 1 to 2 minutes.