Does this apply to parents?
No
I actually think yes. If you’re becoming a parent because you want to rule a child, you’ll probably be a nightmare parent. If you’re becoming a parent because you want to love and raise a child, you’ll be a good one, and at times when you need to “rule” them you’ll do it well.
If the incentives were such that parents wanted to leech off their children, and only have children in the first place in order to create slaves who will do their every bidding, then perhaps. But that's the opposite of how every parent is incentivized, and would effectively destroy the human race in a single generation. Unless the parent is extremely mentally ill (they do exist), they will always desire to put themselves on the line to ensure a better present and future for their children. They'll let their children feed off of them (which all babies are literally meant to do!), if that's what it takes. They'll devote every hour of every day to their needs, until they gradually become independent. It's the exact opposite of someone who desires political power over the lives of others.
So, those who want to parent (rule, guide and lead... goveren) children are inherently different beings than those who want to lead, guide, and govern parts of society? But they are not so different. They, leaders and governors, are often parents themselves. Wanting, or being willing, to governmen children or otherwise is in fact not ipso facto proof or even evidence that the willing are least suited to it. In fact if they were so unsuited to it, the example of their parenting would reflect it and we would indeed have no society left. But, as most governors are parents, the locus of the flaw is not merely shown in the desire to govern. Further we can skew our description of parenting to show the incentives are similar. We are wired to raise our children (in part) precisely so they can serve us later on in our older years. I'm not saying the state or our system is good. This rationale for the major cause of the problem just doesn't pass logical muster as regards human nature. You're observation of incentives has more explanatory power than a blanket view of the nature of humans.