When you run something asynchronously it means it is non-blocking, you execute it without waiting for it to complete and carry on with other things. Parallelism means to run multiple things at the same time, in parallel. Parallelism works well when you can separate tasks into independent pieces of work.
Take for example rendering frames of a 3D animation. To render the animation takes a long time so if you were to launch that render from within your animation editing software you would make sure it was running asynchronously so it didn't lock up your UI and you could continue doing other things. Now, each frame of that animation can also be considered as an individual task. If we have multiple CPUs/Cores or multiple machines available, we can render multiple frames in parallel to speed up the overall workload.
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