Transform.localPosition Script Problem.

This temporarily removes the attached parent:

 void OnMouseDown()
  {
      Transform[] childrenTransforms = GetComponentsInChildren<Transform>();
      for(int i =0;i<parent.childCount;i++)
      {
          childrenTransforms[i].parent = null;
      }
      parent.localScale *=expandAmount;
      parent.localPosition *= localPosition;
      for(int i =0;i<parent.childCount;i++)
      {
          childrenTransforms[i].parent = parent.transform;
      }
  }

 using UnityEngine;
 using System.Collections;
 
 public class Expands : MonoBehaviour {
 
     public Transform parent;
     public float expandAmount = 1.5f;
     public float localPosition = 1f;
 
     void OnMouseDown()
     {
         Vector3 oldpos = parent.position;
 
         parent.localScale *=expandAmount;
         parent.localPosition *= localPosition; //this is a bit weird?
         for(int i =0;i<parent.childCount;i++)
         {
             parent.GetChild(i).localScale =new Vector3(1/parent.localScale.x,1/parent.localScale.y, 1/parent.localScale.z);
             parent.GetChild(i).position -= (parent.position - oldpos);
         }
     }
 }

There you go.

Well, the easiest way to not have child objects move when you move the parent, is by not making them child objects. Are you sure they should be child objects if you are trying to move their parent without moving them?

If I look at your code, I get the sense that you might actually want to reverse the parent-child relation.

That said, there are a few ways to solve your actual problem. Easiest might be to put a script on your child objects that constantly assign it it's 'correct' position - something like

 Vector3 worldPos;
 
 void Start(){
  worldPos = transform.position;
 }
 
 void Update(){
  transform.position = worldPos;
 }

That's an incredibly ugly way to solve it, and completely stupid too - if you need to write a script like that you should just unparent your objects - but afaik it should work. Alternatively, you might try moving each child object in the reverse direction and amount you moved the parent object in your script. I'd give you an example but looking at your code your movement makes no sense. You're...multiplying parent.localPosition with localPosition? Where localPosition is just 1? Umm, then your parent object isn't moving at all? And even if it's not 1, multiplying with a scalar is not how you move a point. The more I look at it the more I am starting to wonder about your object hierarchy. Could you describe it and why it's like that?

From what I see you aren't moving your parent object at all, you are just making it bigger, so you could probably also simply resolve this by removing all your movement-related code from that script...

General Coding Tip: write some comments in your code. Not only does that make it easier for others to read and debug it, I find that it often helps me get a clearer image of what is happening when I write down above each line what exactly it should do. Especially useful when I'm coding something complex or in an unfamiliar language.