How to instruct REST API to include additional data for given resource
Spelling check; as far as REST is concerned, you aren't asking for a given resource with additional data, you are asking for a different resource. There's no particular reason that this new resource needs an identifier related to the existing one.
What are the best practices here?
Think about how you would do it with a web site.
The URI would be created by presenting a form to the user, with input elements to allow the client to specify the values in the query string. In other words, the form is playing the role of a URI Template.
So to give the consumer additional control, you would include a new input control in the form - it might be a free text field, or it might be something like a list control which enumerates specific possible values.
Which query param should I add according to industry standards
There are a couple possibilities.
You could review the list of IANA Link Relations, to see if there is a close match. Technically, link relations aren't query parameters. But what this list does is match spellings with semantics, which is what you really want: is there a common understanding of foobar that you can leverage for your own needs.
Another possibility is to look in resources like schema.org, which again has lots of interesting mappings between spelling and semantics.
I happen to know off the top of my head that the EventStore API uses embed with different values to allow the client to specify that it wants to access resources with richer information; but to the best of my knowledge that choice was arbitrary, not based on any real "industry standard".
