Continuity mistake: When the Flintstones and the Rubbles are sitting at their table in the pizza parlour, Fred asks Betty if she wants some pizza and you can see there is a large pizza in front of Wilma and Betty, but in the very next shot the pizza has gone.
Continuity mistake: At the end of the episode, watch the colour of the time-machine inventor's outfit. You can see it's light blue when he pulls the lever, but when he gets the Flintstones and Rubbles back to the fair and asks them what happened, his outfit is a darker shade of blue. It then changes back to its lighter shade as before.
Continuity mistake: In the scene where Fred and Barney are in the doctor's surgery, the doctor produces a lollipop for Fred. When he produces the lollipop, you can see it's well away from Fred, but as the shot changes the lollipop is much closer to him than it was in the previous shot.