AE: Parent Element to a point of a path?

I want to parent an element (pre-compose) to one of the points on my path.

What I'm trying to achieve is a "bounce line" effect to which, I wish to attach an object. I was wondering if I could set the anchor point on my object to be exactly on this point that I'm showing, then somehow, parent these two.

Ultimately, it should move the same as my bounce line.

Thanks!