First, obviously one can suggest a book that they think "demolishes" Ayn Rand's (or any other author's) views, regardless of whether they think that the person they suggest it to supports Rand or not.
But let's get into the implication and see, if it was true, whether it could be justified.
So, you protest on the implication you think the parent makes that "by attacking the use of a quote to ridicule those who have read Ayn Rand and attempt to present her ideas, I am myself promoting Ayn Rand's ideas".
And you name various fallacies to show why this is not necessarily true.
Well, it might not be a logical conclusion in the formal logic sense, but it's a very probable conclusion given all we know about human behavior, how conversations go, etc -- not to mention your writing style.
In life, as opposed to pure math, we often need, and do, judgement calls based on incomplete data, calls that don't actually follow with 100% logical certainty, but are nevertheless quite clear from the general context. Sure, we might be wrong in some cases. That's still a net positive, if we are correct in most cases. 100% correctness is for the inhabitants of Vulcan.
I don't know what the parent consider you -- he might consider you a fellow anti-Rand person for all I know, and suggested the book because you'd appreciate it being such.
But to my eyes you totally look like a pro-Rand person -- not just because you jumped to criticize the quote (yes, it appealed to emotion -- but people tend to ignore that when it concerns emotions and quotes they are in favor of), but also because of the overall tone of the responses and the fallacitis ("oh, I discovered this blunt tool on the internet, let's hammer every conversation into a inquiry for objective truth").