Place the shredded Jarlsberg cheese in a medium mixing bowl.
Add the mayonnaise and whole milk to the bowl.
Add the diced red onion, finely chopped parsley, fresh lemon juice, salt, black pepper, and paprika.
Stir until all ingredients are fully combined and the cheese is evenly coated.
Taste the dip and adjust the seasoning if necessary.
Transfer the dip to a serving dish, if desired.
Cover the dip tightly with plastic wrap or a fitted lid.
Refrigerate for 1 to 2 hours to allow the flavors to blend.
Before serving, stir the dip once more.
If the dip has thickened after chilling, add a small amount of milk and stir until the desired consistency is reached.
Serve chilled with crackers, pretzels, fresh vegetables, pita chips, or toasted bread.