Line a baking sheet with parchment paper or a silicone baking mat or spray the pan lightly with nonstick baking spray.
In a large bowl, combine the flour, baking powder, baking soda, salt, and sugar. Whisk to blend thoroughly.
With a pastry blender or fingers, work the butter into the dry ingredients until the mixture is crumbly and quite fine.
Stir the parsley into the mixture.
Add the olive oil and buttermilk and stir with a fork just until moistened.
Turn the dough out onto a floured surface and knead 3 or 4 times, just until you have a soft but cohesive dough. Don’t overwork the dough.
Pat the dough into a circle about 3/4-inch in thickness and cut out with biscuit cutters. Cut straight up and down — if you twist, the biscuits could rise unevenly.
Arrange the biscuits on the prepared baking sheet; bake for 12 to 16 minutes, until lightly browned.
If desired, brush the tops with some melted butter while they’re still hot.