In a large nonstick frying pan on medium-high heat, melt 1 tbsp. butter. Add mushrooms, lemon zest, and lemon juice. Cover and simmer 5 minutes to allow juices to accumulate.
Uncover and cook over high heat, stirring often, until juice evaporates and mushrooms brown, about 5 minutes. Add broth and stir to release browned bits in pan. Pour into a small bowl, scraping pan well.
Trim fat from thighs. Working with one thigh at a time, place on a piece of plastic wrap. Cover with another piece. Pound meat with a flat-surface mallet until about 1/4 in. thick. Pull off plastic and season with salt and pepper. Repeat with remaining thighs.
Add 1 tbsp. butter to pan and melt over high heat. Lay chicken pieces in pan (you may have to cook in batches). Cook until edges begin to turn white and bottom is lightly browned, 2 to 4 minutes. Turn over and cook until no longer pink in the center, 6 to 8 minutes total (cut to test), lifting to a platter as cooked. Cut into serving portions and cover with foil to keep warm.
Pour mushrooms into pan and add 1/4 cup parsley. Stir in wine, scraping to release brown bits. Immediately add remaining 2 tbsp. butter and stir rapidly until melted. Drain juice from cooked chicken into pan. Bring to a boil, stirring. Pour over chicken and sprinkle with remaining parsley.