Also, since John clearly seems to think that whenever he and his children have the least sort of difference of opinion, they're clearly trying to revolt against his legitimate authority and usher in an era of chaos where his opinion is never to be expressed, it's easy to see that dealing with a teenager is going to make his being a gutless, vindictive and self-absorbed tin-pot dictator even worse than it already is.
That being said, it seems to me that this strip in particular explains what most frightens them about their children maturing: their fear that somehow or other, their children will be just more people whom they cannot control. We are, after all, dealing with two people who base their belief that they are the greatest victims in the history of the universe on the fact that people don't do what they want them to do when they want it done. Since they cannot have the god-like absolute control over everyone ever that their pathetic vanity and hateful megalomania make them want, they MUST own horses so as to be compensated for being defeated by everyone ever.
What this all means is that the general public is expected to part with money better spent on food and shelter so that they can be reminded that the people who will end up picking Elly's retirement home for her are terrible because they didn't agree that Mommy and Daddy cannot be questioned ever.