Welcome to the Airtable Community, Samuel! I assume that you’re talking about the listing of linked records that you see when you click the + button in a linked records column in a grid view. If I’m right about that, then actually you do have SOME control over the fields that are shown.
Let me approach this in two ways.
1. Make linked record primary field a formula field
First, you control the definition of the primary field in the linked table — the one that says “Mill County Texas Cemeteries (2002)” in the second record in your screenshot. If you need to see more info in this listing in order to pick records efficiently, you might want to consider changing your primary field to a formula field that concatenates the fields you want to see. I do this a lot.
2. Editing hidden fields list in default view of the linked table
The second approach is a little less straightforward and may even be an accident of Airtable’s current deep design that could change in the future.
You know by now that the Primary Field in a table in Airtable — the column that’s fixed to the left of every grid — is very important. Well, the first view in every table is also very important.
I don’t mean the view that is first in the list of views for that table. I mean the default view that was created automatically by Airtable. This view is important even if it has been moved — in fact, even if it has been deleted. It seems to determine how the table is viewed from the perspective of other tables (unless the link between tables is itself defined on the basis of a specific view). [Corrected 14 hrs later per response from @Martin_Kopischke below.]
Now, what matters isn’t how the view itself looks to a user. What matters is the order of the fields in the Hidden Fields menu.
The screenshot above shows the hidden fields list for a Notes table that is a child table linked to another table. When I go to add a note from the other table by clicking the + button in the LinkedNotes column in a grid, what I see is a short version of the note (the primary field) + Created + CreatedBy + Status. NOTE that all three of those fields are in fact invisible in the Notes grid! By moving those fields up and down in the Hidden Fields menu, I can modify the fields that appear in the other table. Here’s the result:
Not entirely satisfactory, but a useful tip in some circumstances.