On most websites I've found, people post MOCs that use a random (or seemingly unlimited) supply of pieces. However, I'm interested in building MOCs using only pieces entirely from one set. (For example, I'm interested in reusing the parts from the 10243 Parisian Restaurant.) Are there websites dedicated to this idea?
Rebrickable may have exactly what you are looking for in addition to a lot of other features. You can search for a set and see alternate models that builders have submitted. Here's the page for the Parisian Restaurant. It shows one submitted MOC using just the parts from this set:
There tend to be more results for smaller sets, particularly in the Creator theme as these have many reusable parts. For example, Sunset Speeder currently has 43 alternate models available:
If you create an account, you can add all of the sets that you own and see which MOCs you are able to build using all of the parts you have available.