But that "sliver" of a problem is known as the hard problem of consciousness for a reason [1], which is exactly the sort of problem neuroscience can only address in a limited capacity. Understanding how nerves propagate a signal to produce a sensory input (an "easy" problem of consciousness) doesn't inform us as to why certain physical mechanisms result in conscious experience (or more fundamentally what it even means to have a conscious experience).
To return to the topic at hand, a stochastic parrot generates grammatical, sensible language without understanding its underlying meaning. Of course, you can debate what it means to understand something; but for a person to vocalize an idea they understand, they must first somehow consciously process that idea. This is firmly a hard problem to which neuroscience offers limited guidance.
Of course, I'd agree that human beings aren't stochastic parrots -- if human beings were stochastic parrots, then what would it even mean to understand something? But I doubt you could use neuroscience to ascertain whether large language models are or aren't stochastic parrots. Indeed, depending on your definition of "understanding", consciousness might not even be a prerequisite, making the comparison to neuroscience moot.
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[1] https://en.wikipedia.org/wiki/Hard_problem_of_consciousness