Flash was notorious for security holes. Since it no longer receives patches, running an old .exe installer exposes your system to malware that specifically targets these unpatched vulnerabilities.
For some standalone desktop applications, Adobe AIR (now managed by HARMAN) still provides a supported environment. Conclusion flashplayer320r0344winaxexe
Adobe hard-coded a "time bomb" into the final versions of Flash. Even if you successfully install version 32.0.0.344, it likely will not play content, displaying a static "EOL" icon instead. Flash was notorious for security holes