I find it easier if I disassociate the English meanings of the words from really how the words are used to mean different things in computer science. To me, concurrency is only in the programmers mind. That is, a programmer writes concurrent programs with different threads of control, say, one to process the user input, one to process the network packets etc. In reality, the multiple threads of control could really be running SEQUENTIALLY. That was the case when there was only one core. Remember, pthreads predates multi-cores.

Parallelism is something more concrete, it REALLY exists in the real world. A programmer could just imagine and write a sequential loop, but if the compiler deemed it to be really parallel, it could really run parts of the loop in parallel. Of course, a programmer could write a concurrent program, as in, multiple threads of control, and the system could really run it in parallel, say on different cores of a multi-core processor. So remember, concurrency is abstract, parallelism is concrete.