Update docstring options parser and add docs cron job (backport #1312)

[mergify]

This is an automatic backport of pull request #1312 done by Mergify.
Cherry-pick of f16be3b has failed:

On branch mergify/bp/stable/0.5/pr-1312
Your branch is up to date with 'origin/stable/0.5'.

You are currently cherry-picking commit f16be3b.
  (fix conflicts and run "git cherry-pick --continue")
  (use "git cherry-pick --skip" to skip this patch)
  (use "git cherry-pick --abort" to cancel the cherry-pick operation)

Changes to be committed:
	modified:   docs/_ext/custom_styles/option_parser.py
	modified:   docs/manuals/characterization/tphi.rst
	modified:   docs/manuals/verification/randomized_benchmarking.rst
	modified:   docs/tutorials/custom_experiment.rst
	modified:   docs/tutorials/getting_started.rst
	modified:   requirements-dev.txt

Unmerged paths:
  (use "git add/rm <file>..." as appropriate to mark resolution)
	both modified:   .github/workflows/cron-staging.yml
	both modified:   docs/tutorials/calibrations.rst
	deleted by us:   requirements-extras.txt
	both modified:   tox.ini

To fix up this pull request, you can check it out locally. See documentation: https://docs.github.com/en/github/collaborating-with-pull-requests/reviewing-changes-in-pull-requests/checking-out-pull-requests-locally

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[coruscating]

Closing in favor of a one-time manual docs redeploy.