One could argue, that given the name "bridle path" they have more "right" than anyone else.
But,..."footpath", "roadway"?
Personally, I've never taken a bridle for a walk, nor it me, although a few have been attached to horses which took me for walks or faster.
Where did "walk" come into it?
"Bridle path" is a path for bridles, (presumably attached to horses). "Footpath" is a path for foot.
"Path": noun: a line or route along which something travels or moves