Salt the water, then add the asparagus and return to a simmer. Reduce the heat to medium and cook for 3 to 5 minutes, until the asparagus is crisp-tender when pierced with the tip of a paring knife.
Using tongs, remove the asparagus from the water and place it in the ice water for about 20 seconds, until fully cooled. Drain and place on a paper towel–lined plate to absorb excess water.
Chop the asparagus and add it to the bowl with the artichoke hearts. Add the avocado, and add lemon juice and additional salt if needed. Top with the parsley and pinenuts, spoon onto plates or bowls, and serve