Bergman's visuals asides
One of the comment contributors has already pointed at one or two little allusions that have nothing to do with what Mozart/Schikaneder wrote. Most of them are in the opera's interval. We see
a) Sarastro reading a book labelled 'Parsifal'; which pans to
b) A boy reading 'Kalle Ankas' (Donald Duck);
c) The Queen of the Night smoking in front of a no-smoking sign;
d) Tamino losing a game of chess to Pamina
a) Parsifal (the book is probably a score) is Wagner's last opera. It uses Wolfram von Eschenbach's medieval tale of Parzifal to tell the story of the redemtption of a devout sect of knights riddled by doubt and beset by the threat of moral degradation. The hero and saviour is, like Tamino, a young man who stumbles into their midst and is guided into his messianic role by a wise old man, Gurnemanz, who would correspond here to Sarastro. It's very good but very long and very serious.
b) Donald Duck is a cartoon about a Duck called Donald who has a lisp and is impossible to take seriously in any context. Bergman is saying 'this opera is terribly serious... just kidding'!
c) In 1791 when the opera was first performed, the Queen, singing a coloratura role (very runny and high notes), would have been instantly recognised by a contemporary audience as a baddie. Indeed, Bergman goes on to point this out in more modern terms later in the opera proper, using the old pantomime staple of green lighting. This smoking vignette may be seen as both a little theatre joke and also a nudge for those who, reasonably, had not yet managed to place the Queen's moral duplicity.
d)... So a) a nod to a high water-mark of German opera during a German opera; b) instantly undermining the pompousness of that; c) a nod to theatre traditions old and new; and finally d) a self-nod to the director of The Seventh Seal, with its famous beachfront chess match.