Let me try to explain the difference between the picture and the video. Btw, lots of people would find BOTH offensive.
The video uses IRONY and OVERSTATEMENT (hyperbole) to create humor. So, the woman offers her opinion, which is clearly reasoned and educated. We'd expect the dinner table to respond positively and go on with the discussion, instead, everyone is offended. The "offense" is that this opinion was rendered by a woman. How inappropriate for a woman to "disrupt" in such a manner. (Clearly, it wasn't disruptive, but pertinent...so the interpretation of it as disruptive is over-reactive and creates IRONY). This is continued with the clearly hyperbolic and unscientific "visual" presented. In other words, the video creates a lot of context to understand that this is meant IRONICALLY and in JEST.
Not so with the picture. While I'm sure people who knew this family got the joke, there just isn't enough information present to provide context. Instead, we just get images of women bound and gagged and the implication of various stereotypes--that women talk too much, that women are disruptive forces that must be contained, etc. It seems especially "ouchy" because little girls are bound and gagged in the picture. There's no larger context to create a deeper understanding, so we're left with this pretty yikes image. While this family was likely attempting the same sort of joke as the video, it ends up being a misfire and coming across as completely sexist.
Ultimately, someone is likely to be offended by both if they don't understand irony. It's like all those people who get up in arms over an internet article only to realize it was from The Onion.