Preheat the oven to 350°F (175°C) and line an 8" by 5" (20cm by 13cm) loaf pan with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt.
In the bowl of a stand mixer fitted with a paddle attachment (or use a hand mixer and a bowl), cream together the sugar, unsalted butter, and vanilla until light and fluffy.
Then, add the oil and beat until has incorporated.
Add the eggs, one at a time, mixing well after each addition.
Add the puréed bananas and grated zucchini and beat on low to incorporate. You can also fold with a rubber spatula.
Gradually add the dry ingredients to the wet ingredients, mixing on the lowest speed until just combined (or fold the dry mixture using a rubber spatula).
Pour the batter into the prepared loaf pan and bake for 50-60 minutes, or until a toothpick inserted into the center of the loaf comes out clean. Cover with a piece of foil if the top browns too much during baking.
Allow the bread to cool in the pan for 10 minutes, then remove it from the pan and transfer it to a wire rack to cool completely before slicing and serving.