I am tempted to call it a simple genetic fallacy, a fallacious assumption that the argument is wrong because of the person making it. But that's not all that's going on here. What's going on is that the person arguing for better parking for mothers is reading a dismissive argument, "they should just walk," and perceiving that it must have come from someone that is perhaps opposed to all procreation, or--less sarcastically--someone who just hasn't fully unpacked all the biases that make them dismissive.
C.S. Lewis coined the term bulverism for this implicit assumption that the flaw in the argument, "they should just walk" is that the argument comes from a biased person AND must be wrong because it comes from a biased person. Both assumptions are fallacious on those grounds.
"You must show that a man is wrong before you start explaining why he is wrong. The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became so silly."
In this case, you'd assume "they should just walk" is a ridiculous, biased thing to say, and then get busy explaining why it's stupid to marginalize the need of mothers to walk shorter distances. The fallacy is skipping right over explaining the need to arguing that the need is obvious and should not be marginalized, and that anyone who marginalizes the need is biased.
Some anonymous genius at Wikipedia contiunues:
Assuming one's opponent is wrong is a formal fallacy of circular reasoning. Undermining one's opponent rather than arguing that he is wrong is a fallacy of relevance or genetic fallacy. Bulverism combines both of these. One not only assumes one's opponents are mistaken but also accuses them [of] believing the mistakes because of their motives or some accidental features of who they are.
In the situation the OP describes, there's bulverism at play, because "they should just walk" is assumed to be risible, and the only criticism of the argument is that it's inherently biased. A better response is to argue the argument comes from someone unaware of the societal benefits engendered by giving parents better parking spaces, whatever those are. You should not argue that it must come from someone who hates breeders.
I happen to be a father of two, and I'd love the occasional better parking spot. But "just" walking is exactly what we do with our children, and I'd find it difficult to explain why any particular parking lot is an exception to all the others. Perhaps full of unvaccinated people and roving bands of dingoes? I don't know.