For the filling: Heat the oil in a large skillet over medium heat. Add the garlic and onion and cook until softened, about 4 minutes. Add the zucchini and yellow squash, season with salt and pepper and cook, stirring frequently, until the squash softens but is not mushy, about 4 minutes. Pour the mixture into a medium bowl and let cool.
Add the ricotta, Parmesan, parsley, egg, 1/2 teaspoon salt and 1/4 teaspoon pepper to the cooled vegetables. Stir to combine.
For the tomato sauce: In a food processor, pulse the tomatoes with their juices until they are coarsely ground or chop the tomatoes by hand. Transfer them to a bowl, add the salt and pepper, then stir in the basil.
For the manicotti: Preheat the oven to 375 degrees F. Grease a 9-by-13-inch baking dish with oil or cooking spray.
Transfer the filling mixture to a large resealable plastic bag. Snip a large hole out of one corner of the bag with scissors and use the bag to pipe the mixture into the cooked manicotti shells.
Spread about 1 cup of the tomato sauce on the bottom of the baking dish. Place the manicotti shells into the baking dish. Spread the remaining sauce over the manicotti. Top with the mozzarella and Parmesan. Bake until the sauce is bubbling and the cheese is melted, 35 to 40 minutes. Let stand 10 minutes before serving.