You can download a Parsec installer from the downloads section, for any of the supported operating systems contained there. Make sure to get the correct 32 or 64-bit version for your system to avoid issues while installing. You can also check the articles for the supported hardware for connecting to a computer or hosting your computer.
If your download isn't finishing, the installer file is disappearing, or the installation is ending prematurely, check if your anti-virus is interfering. A common culprit is Segurazo / Santivirus / Segantivirus, it commonly causes installation issues and most of our users don't even know they have it installed. If you have it, uninstall it completely and restart your computer. In newer variants of Segurazo, you will need to use safe mode and use something like Malwarebytes to remove it.
While installing on Windows, you will have to choose whether you want a "Per User" or "Shared" installation. "Per User" will use different accounts for each user and won't start Parsec until you log into Windows. "Shared" will allow access on the Windows login screen but will also use the same account for all users on the computer. Once you've decided which one makes more sense for you, click Next and Parsec will install.