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I'm not convinced that is a trope, to be honest. It is a thing that happens, certainly, but we call those People Sit On Chairs.
How does her improvised cleaning of the kid's face help to tell the story? Does it have plot importance?
If the child hates when the parent does this in public, it's probably Amazingly Embarrassing Parents.
If the parent who licks the napkin is Terrified of Germs, it can be Hypocritical Humour; otherwise, I would go with Amazingly Embarrassing Parents. It might also be that the "kid" is a teenager or young adult, who should know better.
Someone (usually an overprotective mom) will notice some dirt on the kid's face, pull out a napkin, lick it, and apply it to the kid's face to clean it.