Combine paprika, pepper, and salt in bowl. Pat pork dry with paper towels and season with spice mixture. Heat oil in 12-inch skillet over medium-high heat until just smoking. Brown pork on all sides, about 8 minutes; transfer to plate.
Add 2 tablespoons butter, mushrooms, and shallot to now-empty skillet. Cover and cook, stirring occasionally, until mushrooms release their liquid, about 5 minutes. Uncover and continue to cook until vegetables are lightly browned, about 5 minutes. Stir in thyme and cook until fragrant, about 30 seconds. Stir in broth, scraping up any browned bits, and bring to simmer.
Nestle pork into skillet and add any accumulated juices. Cover, reduce heat to low, and cook until tender, about 15 minutes. Transfer pork to serving platter, tent with aluminum foil, and let rest while finishing sauce.
Bring sauce to simmer over medium-high heat and cook until liquid has reduced to about ¼ cup, about 4 minutes. Off heat, whisk in remaining 2 tablespoons butter, 1 piece at a time, until incorporated. Whisk in vinegar and parsley and season with salt and pepper to taste. Spoon sauce over pork and serve.