Place potatoes in a medium pot and cover with cold water by 1 inch. Season with salt; bring to a boil over medium, then reduce to a rapid simmer. Cook until tender, about 8 minutes. Drain potatoes and return to pot.
Meanwhile, heat oil in a medium skillet over medium-high. Pat steaks dry with a paper towel and season generously with salt and pepper. When oil just begins to smoke, add steaks to skillet. Cook until browned well on all sides, turning with tongs as needed, about 10 minutes total (reduce heat to prevent burning as needed). Reduce heat to medium, top steaks with thyme, and add 1 tablespoon butter to skillet. When butter froths, tilt skillet slightly so butter pools and repeatedly spoon it over steaks, 1 minute. Transfer steaks to a plate and tent with foil.
Briefly rinse skillet under cold water and return to medium heat. Add 1 tablespoon butter to skillet; when it froths, add shrimp and cook until beginning to brown at edges, about 2 minutes, turning halfway through. Season with salt and pepper, add half the parsley, and toss. Transfer to a plate.
Return skillet to medium heat and add cooked potatoes. Cook, stirring, until potatoes are heated through and starting to break down, 1 minute. Add 1 tablespoon butter and remaining parsley and toss to coat. Season potatoes with salt and pepper. Serve steak and shrimp with potatoes.