Preheat oven to 325 degrees F (165 degrees C). Grease and flour 2 8×8-inch baking dishes.
Sift flour, pumpkin pie spice, salt, baking soda, and baking powder together in a large bowl.
Beat eggs, sugar, vegetable oil, pumpkin, and vanilla extract together in a second bowl until mixture is smooth and creamy. Stir wet ingredients into flour mixture until combined; gently fold in zucchini and walnuts. Divide batter into the prepared baking dishes.
Bake in the preheated oven until cakes are lightly browned and a toothpick inserted into the center comes out clean, 40 to 60 minutes. Allow to cool on rack for 20 minutes before removing from pans to finish cooling.