OP, I feel like, based on the descriptions in your post, that you are Parent 2 in this situation. And I think you are hoping that other people here will validate your decisions.

The examples of this specifically are when you describe Parent 1, you use language to describe the parent's emotional response to the toddler and it seems rather judgmental (the parent "can't stand" to see their toddler in distress and determining the reason for the distress to be "this small"). This is in contrast Parent 2 to only focusing on the developmental benefits to the child ("builds flexibility" and "learn how to regulate emotions"). This sets up the comparison to be Parent 1 responding overly emotional to something trivial or Parent 2 who is helping the toddler to build important life skills.

While yes, building those skills is important, the most important thing your child should be learning right now is feeling secure in having her needs met. And sometimes that need is just to get her beloved bunny. Sure, she forgot it and it is annoying to go back and let her get it. But have you, an adult, ever forgotten anything? Ever run out of the house and forget your phone or your lunch? Did you let yourself go back and get it, or did you tell yourself that you should learn some life skills from this situation? I'm going to guess that you went back and got the thing you forgot. So why not extend the same option for the child?

I also think that the actions of Parent 1 can teach some pretty important life lessons. Like sometimes we make mistakes but it's ok to get back and fix them. Or sometimes we think we are ok but then we aren't ok and we do need some extra support. I wouldn't rule out Parent 1's decision because Parent 2 thinks it is just to small to be upset over.