The reason for their refusal, of course, is that like a lot of children, Mike, Liz and April thought that being laughed at for letting their mommy dress them like they was little babies or somethin'was more of a problem than getting cold or wet. As you might have guessed, Elly is pretty much incapable of understanding any of this. Not only does it not make sense to her that her children would like to avoid being made to feel like shit because she's not the one getting teased, it seems to me that she was a preternaturally pliable child growing up. The same kid who gobbled down everything on her plate because she wanted Mommy to love her better than stinky ol'Phil would, I should think, have been dead easy to convince to bundle up because having Mommy love her and tell her that she's right and pretty and smart meant more than what other people said.
That being said, we also have to remember that Elly takes everything too blasted personally and sees everything as being part of her horseshit, fake, super no-way insane fantasy conspiracy involving everyone ever to ruin her life and laugh at her for wanting to be happy too. This means that Mike isn't carrying an umbrella because he doesn't want to be teased, he's doing it to reject all the love in her great big heart. The idea that she's not really a factor in the equation is not one that can occur to someone as vain and silly as she is.