There was a firm that a year or so ago tried as a model that people could regionally rent a domain name in parallel. I can't remember their exact name but it had dyn in it and were on a .co I think. Anyway, after trying some advertising, they decided it was not worthwhile.
Lets say someone had a name like pizza. The business would see where the person typing name was, and you could purchase an area (like I could purchase western Canada). Someone from western Canada could get directed to my site, while someone from Europe a totally different redirect, all from same business. So same name could get redirected in different ways. While obvious possible concerns (abuse, reputation, VPN), the idea seemed interesting. Too bad it did not catch on. I think problem was they did not start with a large enough selection of quality names.
As others already pointed out, conventional renting can be done at DAN, Epik, through Escrow and probably others, or by setting it up privately through a domain attorney agreement.
Some companies also offer third level redirection rental. For example Names.Of.London own the domain name for.sale. I could rent out (if available) domains.for.sale and someone else in parallel tractors.for.sale and a third person cottages.for.sale.
Bob