cast the Parent object to Child object in C#

I do so (this is just an example):

using System.Reflection;

public class DefaultObject
{
    ...
}

public class ExtendedObject : DefaultObject
{
    ....
    public DefaultObject Parent { get; set; }

    public ExtendedObject() {}
    public ExtendedObject(DefaultObject parent)
    {
        Parent = parent;

        foreach (PropertyInfo prop in parent.GetType().GetProperties())
            GetType().GetProperty(prop.Name).SetValue(this, prop.GetValue(parent, null), null);
    }
}

Using:

DefaultObject default = new DefaultObject { /* propery initialization */ };
ExtendedObject extended = new ExtendedObject(default); // now all properties of extended are initialized by values of default properties.
MessageBox.Show(extended.Parent.ToString()); // now you can get reference to parent object

If I understand your "I just want to fill them automatically" comment correctly, you want to create a new Child object that's populated with the values of the Parent, with default values for the new properties. Best way to do that is to create a constructor that copies the values:

public class Parent
{
   public string FirstName {get; set;}
    public string LastName {get; set;}
    public string City {get; set;}
}

public class Child : Parent
{
    public string PhoneNumber {get; set;}
    public string MobileNumber {get; set;}

    public Child (Parent parentToCopy)
    {
        this.FirstName = parentToCopy.FirstName;
        this.LastName = parentToCopy.LastName;
        this.City = parentToCopy.City;

        this.PhoneNumber = string.Empty; // Or any other default.
        this.MobileNumber = string.Empty;
    } 
}

Now you can use LINQ, like the answers above, to create a Child out of each Parent:

List<Child> lstChild = lstParent.Select(parent => new Child(parent)).ToList();

Note that this is very similar to @daryal's answer, but wraps the parent-to-child copying logic inside the constructor, rather than having it outside in the new Child() call.

I did like this:

class Parent
{
  ...
}

class Child :Parent
{
  ...
  public Child(Parent p)
  {
            foreach (FieldInfo prop in  p.GetType().GetFields())
                GetType().GetField(prop.Name).SetValue(this, prop.GetValue( p));

            foreach (PropertyInfo prop in  p.GetType().GetProperties())
                GetType().GetProperty(prop.Name).SetValue(this, prop.GetValue( p, null), null);
  }
}