Preheat oven to 375°F. Grease a 9x13-inch baking dish.
Warm olive oil in a large skillet over medium-high heat.
Sauté onion until translucent, about 4 minutes. Stir in garlic for 30 seconds.
Add squash and zucchini. Cook, stirring, until just tender and most liquid evaporates, 7–9 minutes.
Spread vegetables on a towel-lined sheet to release steam. Cool 5 minutes.
Whisk sour cream, mayonnaise, flour, salt, pepper, thyme, and red pepper flakes in a large bowl.
Fold in 1 cup cheddar and 1/4 cup Parmesan.
Add cooled vegetables to the bowl and toss until evenly coated.
Scrape mixture into the prepared dish. Sprinkle with remaining cheddar.
Combine panko, melted butter, and remaining Parmesan. Scatter evenly over the top.
Bake until bubbling and golden, 22–28 minutes. Rest 10 minutes before serving.
Garnish with parsley and serve warm.