If I install the Visual Editor mediawiki extension on the Wiki that I manage here at my work and also setup Parsoid, will new pages be saved with Wikitext still?
Yes. All content is currently stored as wikitext, so you are naturally able to use the wikitext editor to create pages or edit existing pages.
In the longer term we are working on the ability to store HTML instead for new wikis, in which case it might become possible to run without Parsoid if you don't need a wikitext editor front-end. This is not going to happen over night and will just be an option, so no reason to worry.
So right now, assuming I don't have Parsoid, Visual Editor creates pages in Wikitext, it just can't edit previously existing pages? I would assume that this also means that without Parsoid if I edit a page in the regular Wikitext editor, I won't be able to use Visual Editor with it anymore.
The documentation for the extension's installation mentions
Do you have a link to this documentation?
I was reading this page: https://www.mediawiki.org/wiki/Extension:VisualEditor
Thank you, Derric Atzrott