I'm not sure how this applies? We're talking about the "Claude Code on the Web" custom sandbox, not running Claude Code in GitHub Actions.
Check out Crystal if you want a good tool for managing parallel sessions locally https://github.com/stravu/crystal
You can ask Claude to install Postgres and it should just work. We'll have it. In the default image shortly.
Use pglite
I'm using Claude Code locally a lot, occasionally with a couple parallel session.
I was very happy when they made the GitHub Action - I used it quite a bit, but in practice I got frustrated that I effectively only get a single back-and-forth out of it, I can't really "continue the conversation without losing context" - Sure, I can respond to it in the PR it makes, but that will be a fresh session with a fresh empty context.
So, as much as I don't like moving out of my standard development workflow with my tools, I think this could be quite useful. The ability to interrupt and/or continue a conversation should be very nice.
My main worry is - usually my unit tests and integration tests rely on a postgres database running on the machine, and it's not obvious to me if I can spin that up here?