After Effects, can I parent specific points on a path to a null object?
Use the puppet tool to do this. Place puppet pins on the rectangle and they will allow you to distort the rectangle to follow them. You may need other puppet pins or starch to hold the rest of the rectangle in place. 

Then connect their position to the null using an expression:

thisComp.layer("&lt;the name of the null&gt;").transform.position


or if the nulls are parented to something else:

thisComp.layer("&lt;the name of the null&gt;").toWorld(thisComp.layer("&lt;the name of the null&gt;").transform.anchorPoint)


EDIT Are you asking whether you can attach specific points on a vector path to a null? Not natively, but there is a third party plugin that lets you animate individual path points: http://aescripts.com/bao-mask-avenger/ Fairly pricey though.