Slice the fennel and radishes as thinly as possible on a mandolin or with a knife, reserving the fennel fronds for garnish. Toss together in a large salad bowl with the parsley.
Make the dressing by whisking together the lemon juice and oil. Season to taste with salt and pepper.
Add the dressing to the salad and toss gently to mix. Garnish with the feathery fennel tips and serve.