"Is it just me that's taking the mick? Or are other people waiting for it?"
Doctor Who fans don’t, on the whole, much like our show being parodied. For those of a certain age or or beyond, it probably conjures memories of playground taunts, and of responding to suggestions that “Doctor Who done a poo!” with something along the lines of “You’re stupid. You mean the Doctor! Doctor Who is just the name of the series!”1
Nevertheless, the Blu-ray of the 1977/78 series of Doctor Who contains two Who based sketches from Emu’s Broadcasting Company. So presumably there’s some appetite for such things for the hardcore who buy these (wonderful) things. These two sketches, shown on 12 November 1976 and 14 November 1977 respectively, feature Rod Hull and Emu as Doctor Emu. (Everyone else thinks of Rod and Emu as a gestalt, right?) Doctor Emu travels in a red telephone box, rather than a blue one, and is fighting monstrous dustbins in the English countryside.
In Doctor Who terms, these sketches were transmitted during The Deadly Assasin and Image of the Fendahl, and offer us some sort of insight into how Doctor Who perceived at this point, both within the BBC and by UK culture at large. The do so through the choices made by the people responsible for them, rather something more literal, such as newspaper reviews or audience reports.
For example, the music used on both sketches is radiophonic stock, indicating that the public idea of Doctor Who’s music was still firmly in the arena of musique concréte, despite this being years into Dudley Simpson’s bass clarinet fuelled dominance of Doctor Who’s in-story music. (The second sketch features Pot au Feu by Doctor Who theme realiser Delia Derbyshire, which is very on the nose indeed.)