If a search engine supports a specific Schema.org type (like
Organization) and also supports a specific property that takes another item as value (like
parentOrganization), it would of course parse this nested item, otherwise you couldn’t really speak of "support".
So the question really should be: Which Schema.org types/properties does make the search engine use of, and in which context?
For example, it could be conceivable that a search engine considers only a top-level
Organization item for doing something (e.g., displaying a rich snippet), but not a nested
Organization item. This would be stupid, but that doesn’t mean it couldn’t happen.
Every search engine (and other consumer) has its own rules and quirks. Unless you know about them specifically (in which case you could adapt, if your care about that consumer), you should stick to the official definitions and documentation on http://schema.org/, and you should not limit the expressiveness of your structured data just because there could exist consumers that can’t handle it.
About your example:
I'm concerned that search engines may treat
copyrightHolder Organization only as a property of
WebPage and not as a full, standalone entity (worthy of showing up in search engine results on its own, like company information summaries show up on the side of search results).
Well, it is "only" a property of the
WebPage. Without denoting anything in addition, a consumer only learns that this
Organization is the copyright holder of the web page. I wouldn’t expect a search engine to do anything fancy with this (except, of course, displaying who’s the copyright holder).
If you have a page about a
Thing and an
Organization holds the copyright for this page, the interesting part for displaying rich snippets in general-purpose search engines would typically be the
Thing, not the
If you want to denote that the page represents the
Organization, you could use the
mainEntity property. Using this could make sense on the homepage and the about page of the organization’s official website.
By the way, this
mainEntity property is what you probably want to use instead of
mainContentOfPage in your example (background).