The answer is not as simple as one would assume because it depends on the course and the weather. If the weather cooperates and the turf is healthy, it is not uncommon for golf courses to allow carts to crisscross over any surface, aside from putting greens and tees. However, some facilities ask that carts remain on the fairways whenever possible. This is often due to better drainage in the fairways compared to the rough, and because puddles are easier to see in the fairway after recent rainfall. Conversely, some courses ask that carts remain in the rough rather than the fairways. There could be several reasons why, but one is that the irrigation system lacks coverage and the lower-cut fairway turf is weaker than the rough.
Another point of confusion is where carts should be parked, especially around teeing grounds and putting surfaces. The course-by-course differences are less significant when it comes to parking. Most courses ask that carts stay well away from putting greens and the putting green complex. The same is true of tees and their surrounds. If a cart path is available, it is often best to park your cart on the path and walk to the tee or green. Making a conscious effort to keep all four tires on the path when parking can help prevent worn areas from developing in the turf alongside common parking locations.
The one rule of cart traffic management is that there is no one rule. Every course is different, with different infrastructure to support cart traffic. Weather plays a role, as do soils, topography, drainage, irrigation and traffic levels. Cart rules vary from course to course and should be strictly adhered to at each facility because you may not know the underlying reasons why some courses ask you to drive in areas where another course might not.