Preheat oven to 375°F (190°C).
Halve squash lengthwise and remove seeds. Lightly oil a baking sheet. Lay halves on sheet, flesh side down. Bake for 35 minutes. Let cool 10 minutes.
Using a fork, scrape squash strands and transfer to a large bowl. Add yogurt, Parmesan cheese and oregano. Mix well.