Peel and slice the garlic. Peel and thinly slice the shallot. Use a peeler to shave the summer squash into thin ribbons. Cut the corn kernels off the cob. Pick the parsley leaves from the stems. Drain the roasted red peppers. Add the parsley leaves and roasted red peppers to a small bowl.
Place a large nonstick skillet over medium heat with 1 tbsp olive oil. Once the oil is hot, add the sliced garlic and sliced shallot and cook until softened, about 2 to 3 minutes. Add the corn kernels and a pinch of salt and pepper and cook until charred in places, about 3 to 4 minutes.
Reserve ¼ cup of the corn mixture for garnish. Transfer the remaining corn mixture to a blender or food processor with ¼ tsp salt and the sour cream. Blend, adding water, 1 tbsp at a time, until smooth. Taste, and adjust seasoning with salt and pepper.
Place a small saucepan over high heat with the balsamic vinegar and sugar. Bring to a boil and cook until the sugar dissolves and the sauce coats the back of a spoon, about 1 to 2 minutes (watch carefully, this happens quickly). Transfer to a small bowl to cool.
Set the oven to broil on low. Place flatbreads in the oven, directly on the rack. Bake until slightly crisp, about 3 to 5 minutes. In a large bowl, toss the shaved squash, 1 tsp olive oil, and a pinch of salt and pepper. Spread half of the sweet corn purée on each toasted flatbread and sprinkle the reserved corn on top. Layer on the summer squash ribbons and top with parmesan.
Place flatbreads on a baking sheet and broil until lightly browned on top, about 4 to 6 minutes.
Drizzle the summer squash flatbreads with balsamic glaze and top with parsley and roasted red peppers.