Continuity mistake: When Dick is braking the bus with his feet, lots of sparks come from his shoes. When the shot changes to an outdoor view of the bus, there are no sparks to be seen underneath the bus.
Continuity mistake: Dick removes some lid from the bottom of the bus so he can stick his legs out to reach the road. In the next shot when the bus is flying "Speed"-like there is no hole to be seen.
Continuity mistake: Dick's in the lab talking about new technologies. If you look at the can, its lid pops up twice.
Continuity mistake: In the first sequence, when Victoria Dahl falls from a cliff, several shots show her from above and from the same level as she clings to Dick Steele's hand and starts to fall, then a last shot shows her from below as she wags her arms in an attempt to reach the water safely. In the first shots she is wearing sandals with block heels, whereas in the shot from below she (perhaps a stunt-woman) is wearing flats. (00:03:50 - 00:04:18)
Chosen answer: The lady in the red dress is definitely Dr. Joyce Brothers. Her presence is not a spoof of anything. Just another way to add humor to an already humorous movie.