[trainer] start using training_args.parallel_mode
Following up on https://github.com/huggingface/transformers/pull/8877 which adds `training_args.parallel_mode` to make it easy to comprehend which mode the trainer is running under - this PR deploys the new property in a few places.

@sgugger, have I deployed it as you envisioned it?


Thank you for adding this new property, @sgugger - it has indeed improved the readability!