Heat the oven to 350°F. Spread the walnuts on a small baking sheet and toast until fragrant and a shade darker, 6–8 minutes; cool completely, then coarsely chop.
In a large mixing bowl, whisk together the mayonnaise, lemon juice, granulated sugar, kosher salt, and black pepper until smooth.
Add the apples to the bowl and toss to coat thoroughly so they don’t brown.
Fold in the celery, red seedless grapes, and the cooled chopped nuts until everything is evenly coated.
Cover and chill 15–30 minutes, until the dressing lightly clings and flavors meld while the apples remain crisp.
Arrange the lettuce leaves on a platter, mound the salad on top, and finish with parsley. Serve cold.