The major thing was realizing my wife and I care
way too much about kids getting dirty in play. This came mostly from being overwhelmed by laundry and cleaning up (and sand,
sand f-in everywhere).
The other big thing was, I grew so fixated about my own job and various adult stuff, that I actually kind of hated the playground, and often felt impatient when playing (wanting to get back to million unfinished adult things). I got stuck in a self-centered and rather sad mode of thinking, and with my wife going through her own hardships, we found it too easy to just let the kids do jigsaws or play with bricks, rather than playing with them or taking them somewhere.
The song broke me down, and broke me out of my various mental loops, long enough to realize how deeply unsatisfied I am with how I act as a parent. Not just in the usual, system-2 "I realize my performance is suboptimal, I vow to improve it" way. I actually felt the weight of everything, of how much we, how much growth and happiness I am denying our girls, and it flipped something in me permanently that night. Adult problems were still there, but I suddenly felt I really want to play more, and I absolutely want to take them to playground, which became priority more important than work, and generally made me stress a little less about everything. And so I did stay on the playground the next day, and I loved it, and things really got better and happier for everyone from that day on.
Specific things: we stopped minding the sand that much - which meant we started going to the other playground nearby, that's mostly built inside a big sandbox. Kids love it. And I started to engage with some of the more absurd imaginary stories my older daughter comes up with, to her great delight. And ever since, there was no more "can't do X/Y/Z, daddy needs to finish something for work" - I decided that, in Muffin's words, "this is unacceptable!"[0].
It's hard to put into words. It probably all sounds mundane from the outside, but inside of me, the change was as profound as it was unexpected. Switches flipped, priorities realigned, emotions purged.
This was a big thing for me. But there's a lot of smaller things I took away from Bluey, too, that I feel makes me a better parent. It's truly a special show.
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[0] - https://www.youtube.com/watch?v=RqAlhCUgAVI