Preheat oven to 350 degrees F (175 degrees C). Coat a baking sheet with cooking spray.
Place spaghetti squash onto prepared baking sheet with cut sides down.
Bake until squash is tender, about 45 minutes.
Scrape strands of flesh from the squash with a fork and place in a large bowl. Mix flour and Parmesan cheese into squash strands.
Melt butter in a skillet over medium-high heat. Spoon 1/4 cup of the spaghetti squash mixture into the skillet and pat and press to form squash into a patty about 1 inch thick. Cook until golden brown, about 5 minutes per side. Season with salt and black pepper to taste. Serve patties topped with a dollop of sour cream.