Asynchrony means things happen out of order, interleaved, interrupted, preempted, etc. but could still be just one thing at a time sequentially.
Parallelism means the physical time spent is less that the sum of the total time spent because things happen simultaneously.
in other contexts these words don't describe disjoint sets of things so it's important to clearly define your terms when talking about software.
In ecosystems with good distributed system stories, what this looks like in practice is that concurrency is your (the application developers') problem, and parallelism is the scheduler designer's problem.
Therefore I think this definition makes the most sense in practical terms. Defining concurrency as the superset is a useful construct because you have to deal with the same issues in both cases. And differentiating asynchrony and parallelism makes sense because it changes the trade-off of latency and energy consumption (if the bandwidth is fixed).