Alright I just watched enough of that tutorial, and the author of it is not confused or using the term in a misleading way.
Yes the foot bone is the Parent object, and the head bone is the child.
He is parenting the head bone to the foot bone.
Whenever you use the word parent as a verb like that (parent A to B) it always means A is the child and B is the parent.
It follows the natural selection order, select the child first then select the parent.
You could think of it like a function with two parameters. parent(a,b) The first parameter is the child, then the last parameter is the parent object.
I think the trouble comes from "Parent" being use to describe the controlling object, and the linking relationship. Would it make any more sense if it was "Link A to B"? To me at least it is clear that A will be linked to B, thus B will be controlling A. No different if you use the word parent instead of link.