Consider a language runtime for which every thread takes the same global lock on the interpreter, and therefore prevents two threads running 'simultaneously'. You have concurrency but no parallelism.
Well, I disagree that "concurrency" should be defined as "having multiple threads," but if that's how we want to define it...
then I think you conflated the definitions half way through. Your final example, you state, has neither concurrency nor parallelism.
Actually, by your definition of concurrency, it could be concurrent (put each statement in a separate thread and use locks), though I agree that it can't be parallel; it's inherently sequential.