Factual error: Prince Phillip is boarding the yacht, and seen in the background is a red modern vessel that would not have been around in that era. The vessel shows very modern signs of shipbuilding in its new shape, and modern rescue craft. With satellite domes located on the bridge.
Factual error: As HMS Britannia sails North, there is a disabled symbol on her stern. Pretty sure the RN wasn't interested in disabled accessibility in the 1950's.
Continuity mistake: As Philip paints his parrot, its head can be seen half-painted. When Parker enters the room and the camera switches focus to him, the head can be seen in the background fully-painted. However, when the camera returns to Philip, the parrot's head is again only half-painted. (00:24:42)
Continuity mistake: During the photo shoot, after Tony takes the final shot and tells Margaret to "get dressed", so he can show her his place, she is wearing a different blouse and different earrings when she comes upstairs to his apartment.
Mystery Man - S2-E10
Continuity mistake: When Elizabeth arrives to visit Margaret early in the episode she is bare-headed. When she leaves she is wearing a hat and a neck scarf which she wasn't wearing when she arrived.