Rise of the Cybermen (1) - S2-E8
Other mistake: When the Doctor finds the crystal and blows on it, it glows brightly. While it lights up his face as it should, the backs of his hands light up, apparently from the crystal. This is not possible, as the light comes from in front when the crystal is inside the Doctor's hands. (00:14:55)
Rise of the Cybermen (1) - S2-E8
Other mistake: Alternate Jackie is said to be turning 40. According to a newspaper Mickey finds early in the episode, the parallel universe is in the same year as the main universe. When Jackie's age (she claims to be only turning 39) is announced at the party, Rose does not in any way react as though that's different from the age of main-universe Jackie. The problem is that, given that Rose was 19 when she met the Doctor, and it has been, for main Jackie and Mickey, likely at least two years since the events of "Rose" (remember Rose's one-year accidental disappearance), this means Jackie would have had to have gotten married, and had Rose, before she turned twenty, and been a widow by the time she was twenty. Given that Jackie is from the UK, a first-world country, and that she seems to be in her twenties in "Father's Day", this is highly unlikely.
Suggested correction: "Highly unlikely" is not impossible, there are many cases of this sort of thing happening even in the age of contraceptives. The episode is also set during the 80's, so it's possible that Pete and Jackie were misinformed. There are other TV shows such as Raising Hope or Bump that use this as a major plot point - hell, the very next episode after Fathers Day features an approximately 12-year-old mother in the form of Nancy, the 17-year old mother of antagonist Jamie.
