In a small bowl, with a mixer on low speed, beat cream cheese, Italian parsley, chives, lemon juice, and salt until well blended. Add milk and beat until smooth; if mixture is too stiff, add a little more milk. Spoon into a bowl.
Rinse and thinly slice English cucumber (12 oz.), peel carrots and cut into sticks, and rinse and trim celery and cut into sticks. Arrange vegetables on a platter; serve with lemon-herb dip.