For the eggs: Bring a medium skillet of salted water to a simmer over medium heat; add vinegar. Crack eggs into 4 small dishes. Slip eggs into simmering water, turn off heat, cover, and cook until whites are just firm, 4 minutes. Transfer eggs with a slotted spoon to a bowl of ice water and set aside.
Preheat oven to 200 F. Toast muffins, then spread with some of the butter. Transfer to 2-4 plates; keep warm in oven. Melt remaining butter in a medium skillet over medium heat. Fry bacon until browned, about 5 minutes. Place bacon on muffins; keep warm in oven.
For the hollandaise: Melt butter in a small pan over low heat. Whisk egg yolks, lemon juice, 1 tbsp. water, and salt and pepper to taste in a heavy, nonreactive saucepan until pale yellow; then cook, whisking constantly over medium-low heat, until whisk leaves a trail in eggs, about 8 minutes. Remove pan from heat. Add butter 1 tbsp. at a time, whisking constantly, until sauce thickens; gradually pour in remaining butter, whisking constantly, leaving milky solids behind.
Reheat eggs in a skillet of simmering water for 1 minute; transfer with a slotted spoon to paper towels to drain. To serve, carefully place egg on bacon, and spoon some hollandaise on top. Garnish with parsley, if you like.