"I was convinced that the only thing I wanted to do, ever, was to write novels. However, my parents, both of whom came from impoverished backgrounds and neither of whom had been to college, took the view that my overactive imagination was an amusing personal quirk that would never pay a mortgage, or secure a pension. I know that the irony strikes with the force of a cartoon anvil, now."
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.