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She is saying that what her parents told her was very ironic. She is using "ironic" in the sense of "happening in a manner contrary to what is expected."
Usually, when parents give advice like Rowling's probably gave her, they are being practical. They are telling her she needs to get a good education so that she can get a practical job, and not simply waste her time writing stories (because writing stories won't pay the bills). This is ironic because JK Rowling is now a very wealthy person because of her writing.
In older western cartoons (Particularly Bugs Bunny and his associated Looney Toons), a falling anvil is an object that is commonly used for to denote something heavy, violent, or forceful. She is making the reference to show that she means the comment in a light-hearted way. What her parents said is ironic, but she is not upset at them for feeling the way they did.