Can I run two versions of Visual Studio at the same time

Yes, you can run two versions of Visual Studio side-by-side on the same machine. Each installation of Visual Studio must have a unique combination of major version, edition, and update channel. For example, you can have Visual Studio 2013 installed alongside a more recent version, such as Visual Studio 2022, without any issues. This allows you to continue using your existing Visual Studio 2013 for your family website while also utilizing the latest features and capabilities of a newer version.

When you install a new version, you can specify different installation paths to ensure they do not conflict with each other. Once you have both versions installed, you can configure them to publish to different directories as you mentioned (inetpub\wwwroot\WebSite1 and inetpub\wwwroot\WebSite2). After you are comfortable with the newer version and have migrated your projects, you can then uninstall Visual Studio 2013 if you choose to do so.