FPSController not following parented object

I'm using the First Person Controller prefab supplied by the default Unity Imports. I want to make it so that when the player drops onto a ship, they become parented to it so that they'll move along with it so it does not slip out from beneath them.

The Script I made to do this does indeed parent the entire Player Controller Game Object to the ship, but the Player Controller does not move or rotate with the ship.

Here is the script I made to parent the Controller to the ship when it enters the Ship's trigger:

using UnityEngine; using System.Collections;

public class ParentOnEnter : MonoBehaviour {

 void OnTriggerEnter (Collider other) {
     if (other.gameObject.CompareTag ("Player"))
     {
         other.gameObject.transform.parent = transform;
     }
 }

 void OnTriggerExit (Collider other)
 {
     if (other.gameObject.CompareTag ("Player"))
     {
         other.gameObject.transform.parent = null;
     }
 }

}

In summary, the FPSController DOES get parented to the ship object, but doesn't move accordingly with it. The ship is using a RigidBody component, I don't know if that matters though.

If there's any other information that needs to be supplied to answer the question, I'm happy to try and find it.