Not a concern, really. The head would be fully enclosed in areas where it wasn't needed. The rest is about appropriate context, dress, and deportment.

There are parks, and then there are parks. For example, a major city within driving distance has one of the longest continuous, urban river valley park systems in North America. It's a deep river valley, with ravines and plenty of slippery and high-angle terrain to play in, goat paths and game trails galore, and a surprising amount of privacy. You can scramble half a day and see very few people; and the ones you meet will "get" your "walking stick" as a handy item. (I typically avoid areas that are paved and baby stroller friendly.)