Lions, tigers and bears...
OH MY!!!!
I'm sorry, I couldn't resist!!!

Seriously, I have most certainly taken my babies into places where the signs clearly said they were not allowed, i.e. clothing stores, shoe stores, etc. However, the park has rules set for what I would believe to be public safety. With that said, I would not risk my furbaby's safety, let alone my own and fellow family members in this case. I speak from experience when I say the presence of these little darlings can certainly stir up a bunch of hawks over head. So, I can only imagine what ferrocious (sp?) creature would appear should they happen to sniff out your little one during your nature walk...
Have you considered the possibility of having your furbaby stay with friends or neighbors? It's just a thought...