Acting for Two: Jonathan Hyde played both Alan's father and Van Pelt. A common theory is that Jumanji conjures up Van Pelt by taking the form of the person you had dreaded the most, and for Alan, it was his father - especially since Van Pelt spouts out the same criticism as his father.
Defictionalization: A Jumanji board game was released when the movie came out. It shared many elements with the game in the movie, aside from the supernatural elements (like sucking-in-small-children) ... at least as far as we know. It also used cards, not even attempting to copy the magic-8 ball style interface. It came full-circle again when Zathura came out, which used cards.
Reality Subtext: The gun shop owner asking Van Pelt if he's a postal worker. Around the time of the movie, there had been a lot of news stories about postal workers going crazy and shooting up their workplaces, from which we get the expression "Going Postal."
Technology Marches On: Referenced when they start playing the game in 1969, with Sarah remarking that the self-moving pieces must be done with magnets. When the kids continue the game in 1995, Judy suggests it's done with microchips, which actually makes less sense. Plus, Carl proudly showing off his prototype for a modern basketball shoe.