Slice spaghetti squash lengthwise and scoop out seeds. Place cut-side down in a baking dish; add 1/2 inch water.
Bake in the preheated oven until fork tender, about 35 minutes. Cool squash until easily handled.
Shred spaghetti squash from rind using a fork; cool to room temperature.
Combine feta cheese, red bell peppers, sun-dried tomatoes, Kalamata olives, basil, parsley, white onion, and jalapeno pepper together in a large bowl. Stir in Greek salad dressing, balsamic vinegar, white wine vinegar, olive oil, thyme, garlic, ground black pepper, and salt. Add spaghetti squash; toss to combine.