The Parent Constraint tool replaces the native parenting of After Effects and allows to animate those links.
Its use is very simple: just select a layer and click on the Parent Link button.
In the effects of the layer, you can set its parent, and animate a weight if needed. You can duplicate the effect to link the layer to other layers as well.
You won’t see the effect of the parent constraint unless there already are some keyframes, the tool needs actual animation to compute the constraint.
The transformations of the child layer are computed depending on the weights and the transformations of parent layers from the beginning of the composition. When adding a new parent link, you may set a first keyframe on its weight to 0% and only raise the weight when you need the child to follow the parent.