I assume that by doing this if someone types /city it will automatically redirect to /services/city

Not unless you add a suitable 301 redirection to your site’s .htacess file – ideally before the BEGIN WORDPRESS block. There are online ‘htacess creators that can help you create the correct redirect rule.

Also, I read somewhere that if you add a parent page to an already published page, it will break existing links to that page unless you use a redirect plugin or other 301 setting

That would be correct, yes.

I currently have many pages with parent pages that automatically redirect.

Is it possible that this was an inbuilt feature somehow automatically setup when I installed wordpress and built the website?

I just don’t recall setting up any redirects manually.

I currently have many pages with parent pages that automatically redirect.

Redirect in what way? Can you give us an example? Are you currently using a redirection plugin? WordPress core does not incorporate any option to redirect urls automatically if you move or otherwise rename a page or post.