When invoking the "Run VI" method on a VI, reentrancy is disabled so there is only one instance of the VI running. If you check the error ouputs of the "Run VI" method, you will see that only the first call returns no error; subsequent calls return an error indicating that the VI is already running.
Reentrancy works only when opening a strict VI reference and invoking the "Call VI" method, which can't be set to return immediatly; the diagram has to wait until it returns.
The solution is to save the "Run Me.vi" as "Run Me.vit" to make it a template. When a reference is opened to a template, a new instance is created in memory that can run independently of the other.