I have asked permission from two Park & Rec departments and they both LOVED the idea. I also let them know where the cache was placed so that they could let me know if there was any problem with the location, and so that if park workers stumbled upon it, they would know what it was.
You are best to ask permission since some parks allow and some don't. From what I've heard, Washington State doesn't allow them in the state parks, but I bet there are still some around