Preheat oven to 200°. Toast English muffins, then transfer to two plates and keep warm in oven. Brown bacon in a small frying pan over medium heat, turning once, about 4 minutes total. Put 1 slice on each muffin half and return to oven. Poach 4 eggs (go to Sunset.com to search for "perfect poached eggs").
Melt butter over low heat in a small saucepan. Meanwhile, whisk together yogurt, remaining egg, mayonnaise, lemon juice, and salt in a small bowl. Gradually whisk yogurt mixture into butter and heat until just warmed, whisking often, about 25 seconds (be careful not to overcook the sauce, as it will curdle).
Top muffins with poached eggs, pour some sauce over them, and serve any extra sauce on the side. Sprinkle with pepper and parsley.