If this is the reason, and it may well be, I think it's almost unbelievable that this bug in the installer is still there. When you uninstall the old version it takes a lot of registry entries down with it, including many used by the new version. File associations for one thing, and apparently also links to Lightroom. It turns up in the PS forum regularly, so it's not as if it isn't a known issue.
I first encountered this with CS3 and IIRC I had to uninstall, then use a registry cleaner, then reinstall, to be able to move on. But possibly a reinstall over itself - first CS6 then Lightroom - could be enough.
Anyone contemplating removing old and presumably no longer needed CS versions: Don't. Just let them pile up until reformat-time. License-wise, it doesn't matter. If you're dead determined, uninstall the old version before installing the new.