Hm, I think running on a single core is the exact definition of what the "pedants" say is not parallel. If all you have is one core then you can't achieve parallelism under their definition.
I think the terminology is pretty well established now. But I do agree with you that it's a bad choice of words and that it's annoying, intelligent even, for people to pick a particular unintuitive definition and then go around brow-beating people for not using/understanding their definition.
If you're using these as technical terms which have specific technical definitions, then concurrency and parallelism are distinct but related concepts, and parallel computing means executing code simultaneously on separate execution units, not time sliced on a single core. So yes I am actually right about this. Parallel computing is defined this way in CS and it's not a matter of opinion.
I mean it's not even right. Most parallel/concurrent pedants would consider multithreaded code to be "parallel" even if it is running on a single core.
I think the best thing is to talk about threads, because then you can distinguish e.g. OS threads and hardware threads.