Paternalism

I wonder what logicallyfallacious.com called for the paternalism fallacy 

Paternalism: A serious fallacy of ethos, arbitrarily tut-tutting, dismissing or ignoring another's arguments or concerns as "childish" or "immature;" taking a condescending attitude of superiority toward opposing standpoints or toward opponents themselves. E.g., "Your argument against the war is so infantile. Try approaching the issue like an adult for a change," "I don't argue with children," or "Somebody has to be the grownup in the room, and it might as well be me. Here's why you're wrong..."  Also refers to the sexist fallacy of dismissing a woman's argument because she is a woman, e.g., "Oh, it must be that time of the month, eh?" See also "Ad Hominem Argument" and "Tone Policing."

Source: https://utminers.utep.edu/omwilliamson/ENGL1311/fallacies.htm

This is one of those fallacies that is more of a form of rhetoric. It would fall in the category of avoiding the issue . There can be hundreds of reasons why someone would refuse to engage, some of which are justified and some are not. If a 3-year-old child "argued" that unless I buy him the toy, then I am a poo poo face, and I pulled the fallacy of "paternalism,"  would I be unreasonable? Would be unjustified? How about of the same argument came from an adult? Hopefully, you can see that there does exist a class of arguments that are so infantile that they do not justify a response.

Let's look at the response: "Somebody has to be the grownup in the room, and it might as well be me. Here's why you're wrong..." This would simply be an insult followed by what appears to be addressing the argument. Besides being unnecessarily confrontational, there is no problem with this. In this case, the argument isn't being avoided... it is actually being addressed. Now we can look in the answer as to "why they are wrong" for fallacies, but until a reason is given we can't look for a fallacy (error in reasoning).