Bring broth to a boil in a medium saucepan. Meanwhile, rinse and drain rice in a fine mesh sieve until water runs clear.
Add rice to saucepan with broth, bring to a boil, cover and reduce heat to a simmer. Cook until rice is tender and bloomed, 40–50 minutes. Remove lid, fluff with a fork and let any excess liquid evaporate out.
Meanwhile, melt 3 tablespoons butter in a large sauté pan over medium-high heat until foaming has subsided. Add mushrooms, toss with butter, and cook, without moving, until golden brown on the bottom, about 4 minutes.
Stir mushrooms, and continue to cook until golden brown on both sides, about 10 minutes more; stirring twice. Season with salt and pepper and cook 2 minutes more. Transfer mushrooms to a bowl and return pan to heat.
Add oil to pan and heat over medium just until shimmering. Add celery and shallot, season with salt and pepper, and cook until shallot is starting to brown, about 3 minutes. Stir in pecans and continue to cook until pecans begin to toast, 5 minutes. Stir in sage and garlic and cook 1 minute.
Add cooked wild rice, cooked mushrooms, cranberries, and vinegar; cook just to heat through.
Off heat stir in remaining tablespoon of butter and parsley; season with salt, pepper and additional vinegar to taste.