Preheat the oven to 375°F (190°C) and line a rimmed baking sheet with parchment paper.
In a large mixing bowl, whisk together the all-purpose flour, baking powder, salt, and baking soda until evenly combined.
Cut the cold, cubed unsalted butter into the dry ingredients with a pastry blender (or two forks) until the mixture resembles coarse peas.
Stir in the shredded cheddar cheese and dried parsley until distributed through the flour mixture.
Add the buttermilk and 2 tablespoons of the water; mix and knead gently just until the dough comes together—do not overwork.
Turn the dough onto a lightly floured surface and pat it into an 8-inch square about 1-inch thick.
Cut the square into 2-inch squares, then cut each square diagonally to make triangles; transfer the scones to the prepared baking sheet, spacing them apart.
Whisk the egg with the remaining 1 tablespoon of water and brush the mixture over the tops of the scones.
Bake for about 25 minutes, or until the scones are golden brown on top; remove from the oven and cool slightly before serving.