For Parallaxing the background, we need to follow few steps:

1. The type of image using for parallax must be of type texture 
2. Set it's wrap mode to be repeated.
3. Assign the image to a quad 
4. set shader as Uint/Texture
5. Assign the script BackgroundScroller.cs to quad

using UnityEngine;
using System.Collections;

public class BackGroundScroller : MonoBehaviour {

    public static BackGroundScroller current;
    float pos;
    public float speed=0;
    // Use this for initialization
    void Start () {
        current = this;
    }
    
    // Update is called once per frame
    public void go (string dir) {
    
        pos += speed;

        if (pos > 1f)
            pos -= 1f;
        if(dir == "Right")
            this.GetComponent<Renderer> ().material.mainTextureOffset = new Vector2 (pos, 0);
        else
            this.GetComponent<Renderer> ().material.mainTextureOffset = new Vector2 (-pos, 0);

    }

}

7.Create another script Movement.cs and assign it to the camera.

using UnityEngine;
using System.Collections;

public class Movement : MonoBehaviour {
    public float speed=2f;
    //public GameObject target;


    // Use this for initialization
    void Start () {
    
    }

    // Update is called once per frame
    void Update () {
        float _horizontal=Input.GetAxis("Horizontal");

        if (_horizontal < 0) {
            this.transform.Translate (Vector2.left * Time.deltaTime * speed);
            BackGroundScroller.current.go("Left");

        } else if (_horizontal > 0) {
            this.transform.Translate (-Vector2.left * Time.deltaTime * speed);
            BackGroundScroller.current.go("Right");
        }


    }

}

8.Made Quad as the child of Main Camera.

Now As you go Right and Left via arrow keys you can see parallax being happening with the image.

Note: For Reference Please finds the attach unity package with the blog.

 Thanks, Please leave the comments if you have any query.