In a link a parent transfers it's position, size, rotation to it's children, but not the other way 'round.

In this example the little cage is the parent of the cones and it only transfers it's size, which you can set in the Attributes dialog.

The cones are shrinking/growing from their bottom, because their origin handle was moved to the cones base. Scaling takes place in relation to the origin handle, so moving the origin handle often makes a big difference! If you would like to keep an object attached at a specific place, then it's normally a good idea to keep it's origin handle at the linking position.