Preheat the oven to 400°F. Line a baking sheet with aluminum foil.
In a large zip-top bag, combine 1/2 cup Italian bread crumbs, 2 tablespoons Ranch seasoning, 1/2 teaspoon paprika, and 1/4 teaspoon black pepper. Seal the bag and shake to mix the dry ingredients.
Add 4 medium russet potatoes (peeled and cut into 3/4-inch pieces) to the bag. Seal and shake until the potato pieces are evenly coated with the dry mixture.
Add 2 tablespoons melted butter, 2 tablespoons vegetable oil, and 1/4 cup grated Parmesan cheese to the bag. Seal and shake again until the potatoes are evenly coated with the butter/oil and cheese.
Pour the coated potatoes onto the prepared baking sheet and spread them into a single, even layer.
Bake for 15 minutes. Remove the pan, stir or turn the potato pieces to redistribute the coating, then return the pan to the oven.
Bake another 15 minutes, or until the potatoes are golden and tender when pierced with a fork.
Remove from the oven and let cool briefly, then transfer to a serving dish.