Scald grapevine leaves in boiling water in a medium saucepan; drain well. Cut large leaves in half and leave small leaves intact. Set aside.
Sauté onion in 1/4 cup olive oil in a small Dutch oven until tender. Add green onion, rice, parsley, pine nuts, 1/4 cup water, 1 tablespoon lemon juice, mint, 1 teaspoon salt, 1/2 teaspoon pepper, and dill weed; stir well. Cover and cook over medium low heat 10 minutes, stirring occasionally. Remove from heat; cool slightly.
Arrange parsley stalks to cover bottom of a 4-quart Dutch oven; set aside.
Spoon about 1 heaping teaspoon of onion mixture into center of each prepared grapevine leaf. Fold edges of leaf around mixture, overlapping edges to secure. Place each stuffed grapevine leaf, seam side down, over parsley stalks in Dutch oven. Pour remaining olive oil, boiling water, and 1 tablespoon lemon juice over top.
Invert a heavy plate over stuffed grapevine leaves to secure stuffed leaves during simmering process. Cover and cook over low heat 45 minutes; add more water, if necessary.
Remove from heat. Transfer stuffed grapevine leaves to a serving platter; add salt and pepper to taste. Sprinkle with additional lemon juice and garnish with lemon slices to serve.