Combine 1-1/2 teaspoons salt and lemon juice in large bowl.
Working quickly, peel and quarter the celery root. Shred in food processor. Combine with lemon juice and salt, tossing to coat well. Let sit about 20 minutes. The lemon juice will tenderize the celery root, and keep it from browning as you make the dressing.
Bring about a cup of water to boil in medium saucepan; set a medium bowl over pot (to make a double boiler).
Add mustard and boiling water to the bowl, whisking till well combined.
Drizzle olive oil into mustard mixture, whisking all the while. Do this slowly and whisk well so the dressing doesn't "break".
Dribble vinegar into mustard mixture, still whisking constantly; add salt.
Remove dressing from flame and let cool slightly.
While dressing cools, rinse shredded celery root in a fine strainer. Squeeze out any excess moisture and return to large bowl (dry out the bowl first, though).
Toss celery root with dressing till well coated. Fold in mayonnaise gently. Stir in parsley.
This keeps in the fridge for 2-3 days, and gets more and more tender as it sits.