n. Rule or domination by children; the belief that children's needs and preferences take precedence over those of their parents or other adults.
And I realised, once again, we're living in a perilous new world that says kids, no matter what they do, are always lovely and are always to be encouraged — a new world order where the 'needs' of children rule above all else. So much so, in fact, it now has a name, newly coined in the US: The Kindergarchy. …

In these misguided days of the Kindergarchy, we 'aunties' and 'uncles' must urgently rise up and take far more seriously our responsibilities as increasingly isolated champions of truth in a world of parental illusion. And just the other week, a pretty good start was made.
—Sylvia Patterson, “Let's put some misery back in our children's lives,” Sunday Tribune, July 13, 2008
So it's left to non-parents like Irvine Welsh to tell the uncomfortable truths about parenthood: Parents say, "This is great, it's the best thing that has ever happened." But you see them completely eroded by it at the same time.

SO WHAT is to be done?

How can we depose the kindergarchy and get back to a sensible balance between the needs of children and their parents, so we are not completely eroded by the experience?
—Tom Sykes, “Kindergarchy,” Daily Mail (London), July 01, 2008
2007 (earliest)
For some time in America we have, of course, been living under Kindergarchy, or rule by children. If children do not precisely rule us, then certainly all efforts, in families where the smallish creatures still roam, are directed to relieving their boredom if not (hope against hope) actually pleasing them.

Let us be thankful that Thanksgiving has not yet fallen to the Kindergarchy, as has just about every other holiday on the calendar, with the possible exceptions of Yom Kippur and Ramadan. Thanksgiving is not about children. It remains resolutely an adult holiday about grown-up food and drink and football.
—Joseph Epstein, “'Let All Your Thinks Be Thanks',” The Wall Street Journal, November 22, 2007
This fun word is a blend of kinder, "children," and the suffix -archy, "rule."