A delightful Orthodox family, with 9 children, moved into my neighborhood. I introduced myself and let the children play with my dogs over several months... When Yom Kippur came around, the children said they could not pet my dogs. Why would kindness to 2 of God’s creatures not be permitted on the holiest day of the year (I am a reform Jew)? What tenet of Orthodox Judaism would dictate this?

It’s good that you asked, because there is a misunderstanding here. The issue has nothing to do with kindness, for the Torah, already related positively to the needs and even feelings of animals 3,300 years ago, more than 3,000 years ahead of the rest of mankind (e.g. Vayikra 22, 28; Dvarim 22, 7; 25, 4)! Just that there’s a difference of opinion among halachic authorities, whether one is allowed to pet a dog on Shabbat, whether it’s considered muktzeh or not (for brevity’s sake: please check out “muktzeh” on the web), but there’s no difference between Yom Kippur and any regular Shabbat (Saturday). It’s possible that the father just wanted his children to go to shul (the Yom Kippur prayer takes up most of the day) and occupy themselves with the special day, and not pass the time with less spiritual issues, but for that you can ask him what where his intentions...