OOP in VB.NET - Child, Parent, Parents

I'm trying to do something here with VB that I guess I'm not understanding how to do it exactly. Sorry I'm not that good at OOP.

I have a number of things I'm creating and they have two values - parent name and child name (yes, actual people!).

So it would be like this:

Public Class Child
    Public Property ParentName As String
    Public Property ChildName As String
End Class

And then:

Public Class Parent
    Public Property ParentName As String
    Public Property ChildName() As String
End Class

Then I need to add these to a Parents class where a Parent can have one or more children.

I start by adding a Child. If that child's Parent name already exists, just add the Child's name to that parent, but if it doesn't exist, create a new parent (with that child). Then add all parents to a collection of parents (with their 1 or more children).

A resulting list would look something like this:

Parents:

1. Parent: Jonathan Murry
   2. Child: Carl Murry
2. Parent: Kathleen Anderson
   4. Child: Steven Anderson
   5. Child: Deborah Anderson
   6. Child: Thomas Anderson
3. Parent: Xu Jing
   8. Child: Liu Ming
   9. Child: Liu Ning

(note on the last one - the parent/child last names don't need to match - in this case, the children take the father's last name instead of the mother's, but we don't list the father).

How would I create these type of classes so that I can add children to a parent, add a parent to parents and then ensure it is querable with something like Linq?

Thx in advance.