I've had many domesticated rats as pets, and they are actually very clean and friendly animals.
It concerns me that you noticed a fairly bad odor from just walking past the woman--if it even WAS the rat causing the smell, it would have either been using her shirt as a toilet for a month straight, or it wasn't being kept in the greatest or cleanest living conditions. Both are frightening situations!
Domesticated rats are leaps and bounds above their wild counterparts--they are some of the most friendly and social small pets you can have (and without all the nuisance and diseases). They're fairly smart, too--I've had mine trained to recognise their names and come when called, one even played fetch!
As for being in her hair, one thing my pet rats enjoyed doing was perching on my shoulder as I did my daily household things--most of them love human interaction and contact. They'll get tired of the view and move around sometimes.
I don't really see how the animal was allowed in the park, though, especially in the situation described. Weird!