Heat the oil in a large skillet over medium-high heat until hot. Add the meatballs and cook until browned on all sides, about 5 minutes. Remove the meatballs from the skillet to a plate, reserving the drippings in the skillet. Reduce the heat to medium.
Add the butter to the drippings in the skillet and heat until melted. Add the garlic and cook until tender and fragrant, about 30 seconds.
Whisk in the flour and cook until the mixture is foamy and begins to smell nutty. Gradually whisk in the chicken broth and lemon juice and cook until the sauce is thickened, about 2 minutes.
Stir in the capers and meatballs and season with salt and pepper to taste. Stir to coat well.
Divide the meatballs evenly between serving plates and sprinkle with Parmesan cheese and parsley.