OK, having things enter the public domain would be better but if it were possible to pay a certain amount for every X pageviews of a Mario "cover version" and that makes it legit.. that could be pretty interesting and allow for things like this to live on safely via donations or sponsorship.
I can't blame them, because Mario is one the main drivers for sales of Nintedo consoles in the last decades.
Weird Al does song parodies, not covers. AFAIK the prevailing opinion is that his songs would be legally protected under fair use (whether released commercially or for free) but he typically gets permission from the artists as a courtesy.
In a nutshell, cover is treated (it seems) like a reproduction. For example chopin's music is now royalty free and accessible for any purpose, but you need a reproduction rights from whoever has performed that.
I am sure something is lost here in translation since english is not my first language, but I think you can get a clear idea from it.
If that still holds true if the cover is used in a documentary is a separate question. Also note that sampling is not covered by that paragraph (see Metall auf Metall).
So you could create a clone of Monopoly which plays identically so long as you draw your own board, design your own pieces write the wording of rules yourself, give it your own name and so on.
I suspect the issue here is that it looks (and sounds?) identical.