Verdict: Partly
Partly. Parsec make a Linux app, but it can only connect to other machines. Hosting is listed for Windows 10 and later and macOS 10.15 and later, so your Linux PC cannot be the one being shared.
Parsec's download page lists Linux for Ubuntu 22.04 desktop alongside Windows, Mac, Android and the browser. The hosting note next to those downloads is the part that matters: hosting is available for Windows and macOS only.
So the direction decides whether this works for you. Connecting from your Linux machine to a Windows PC at home or at work is supported and works well. Letting someone else connect to your Linux machine, or playing a game that runs on your Linux machine from elsewhere, is not something Parsec do.
Your account and your friends list are unchanged, and the Linux app signs in the same way.
If you need the other direction, Moonlight covers streaming games from a Windows PC to a Linux machine on your own network, and ordinary remote desktop tools like AnyDesk and Remmina handle the office use case.
Streams games from a Windows gaming PC on your own network to your Linux machine. Free, and needs streaming software running on the Windows side.
Your files: Nothing to move. The game stays on the other machine.
For remote desktop rather than gaming. Works in both directions on Linux, unlike Parsec.
Your files: File transfer between the two machines is built in.
Free and open source, already installed on many Linux systems. Fine for work machines, not for games.
Your files: Connects to existing remote desktop servers, with no separate account needed.
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