Combine 2 cups vegetable broth and quinoa in a saucepan. Bring to a boil. Reduce heat to low and simmer, covered, until quinoa is tender, about 20 minutes.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering. Add cremini mushrooms and cook, stirring rarely, until browned, about 4 minutes. Add 1 tablespoon white wine and let it cook down with the mushrooms, about 3 minutes. Transfer mushrooms to a bowl.
Reduce heat to medium and heat 1 tablespoon olive oil in the skillet. Add onions and paprika. Cook, stirring occasionally, until the onion has softened and browned, about 6 minutes. Sprinkle with flour and cook, stirring occasionally, until the raw flavor has cooked off, 1 to 2 minutes. Add kale and garlic; season with salt and pepper. Cook until kale is just wilted, about 1 minute.
Stir mushrooms and their juices back into the skillet. Add remaining 5 tablespoons white wine and cook until wine is almost evaporated, about 3 minutes. Reduce heat to low; add remaining 1/2 cup vegetable broth. Season with salt and pepper. Cook until sauce is slightly thickened, scraping up any bits that have accumulated at the bottom of the skillet, 5 to 7 minutes. Stir in butter until completely melted.
Remove skillet from heat and stir in sour cream; season with salt and pepper. Distribute quinoa between 2 bowls and spoon kale-mushroom sauce on top. Garnish with parsley.