In my experience, being a parent is unfortunately a poor predictor of caring and being ready to act on climate change or its effects.
The split seems mostly generational; my parents and other parents around their age seem to not really care (there are still dinner table debates about whether Al Gore's An Inconvenient Truth was a scam or not), but as my other-millennial friends are becoming parents, the vast majority of them do care, because this issue has been so front-of-mind for people my age. We're hyperaware of the fact that my country has more wildfires, milder winters, and hotter summers than it did when we all were growing up.
(There may be selection bias at play here; while I don't go out of my way to avoid them, I don't have many friends who deny the issue of climate change).
Many of us are also tracking the second- and third-order effects of these things; crop failures, droughts, and the migration and public services strain that come as a result of this; most frustrating is that the biggest voter blocs won't enact policy that's responsive to this, so when the inevitable (mass, climate-change-driven migration) grows over the coming decades, we'll be unprepared, and we'll either be overwhelmed by the movement of people, or complicit in one of the greatest humanitarian crises in history (and very possibly both).