I would probably make sense if the structure was composed of individual modules, and a thread of some super-strong material that strings them together. Creating a tube with a diameter of 2 AU in one go, after all, seems prohibitively difficult. It would be easier to create independent modules, say the size of a world or a moon, then spin a giant piece of thread around the Sun, and attach them. This allows the tubeworld to grow incrementally over time, starting with just a relative handful of modules attached to a naked string, and then gradually filling out the ring. This allows older generation modules mixed with newer generation modules, and if we assume it's a polysocietal or even multi-species construct, then you can have modules in completely different styles and with different living environments.
The idea isn't bad, I like it, what do you mean by: "modules in completely different styles and with different living environments"?
One way to think of it as a train with no end. The modules would be the cars, though of course the cars might be the size of worlds. There could be ancient modules, and newer ones in newer styles and based on newer technologies. The disparity could be immense, since we're potentially talking about deep time, not just a few years, centuries, or even millennia. Then consider modules added by different civilizations. They might have very different needs and styles. Or species -- there could be water modules, or near-vacuums, and so on. Some modules might have shared environments with their neighbors, with the "doors" between the modules open.
Others might be closed, because of incompatible environments, or worries of ecological contamination. It would still make sense to have a standard common ground, so travelers could pass down the torus, but the common ground is most likely a contained vacuum. Say a torus within a torus, an internal tube with no impeding air. That would allow transit using mono-ships, that are propelled magnetically or gravitically, and which could use gravity differentials or centrifugal force to reach ridiculous speeds, like particles in a collider.
Incidentally, you don't have to "park" it anywhere. Since a stellar engine is based on sublight travel, perhaps very slow sublight travel with transit times in the millions of years, it could be on its way to its next stop and still approachable or visitable. Not dissimilar to the Puppeteer's Fleet of Worlds, except much grander in scope.