I am a bit confused about the words parallel and parallelism: When and where should one use which?
First of all, consider the grammatical class they belong to in order to understand usage better:
When it's an adjective, as you may already know, it's used to classify objects that put side by side have the same distance continuously everywhere.
But it has also other meanings: about Computing, for example, it means "involving the simultaneous performance of operations". It can be also related to Music or Grammar.
As F'x said, it means: "the state of being parallel
So, if you're discussing mathematics, computer science or literature, odds are you want parallelism. For everything else, parallel is more likely what you need.