Bring a saucepan of salted water to a boil. Fill a bowl with ice water. Cut off 1 inch of the asparagus tips and cut each tip in half lengthwise. Using a peeler or mandoline, shave the asparagus spears lengthwise. Blanch the asparagus tips and strands in the boiling water until just heated through, 10 seconds. Drain and transfer to the ice water to chill; drain well. Pat dry and transfer to a bowl.
In a blender, blend the oil, lemon juice, basil, shallot and 1/4 cup of the parsley. Season the dressing with salt and pepper.
Add the pea shoots, pistachios and the remaining 1/4 cup of parsley leaves to the asparagus. Toss with half of the dressing and season with salt and pepper. Add the crab, avocado and the remaining dressing, toss gently and serve at once.