Most web applications (every one I've ever worked on) use connection pooling to run multiple transactions in parallel. I suppose you could think of that as a sort of network level parallelism, but it's not multithreading.
Connection pooling is of course not without it's hazards, scaling databases can be very difficult and almost all of the production incidents I've dealt with involve a database running out of a resource (often connections). But for your garden variety web app, it certainly isn't a dichotomy between serializing all concurrent updates or losing atomicity.
