The problem is, psexec requires you to accept a license / eula. If you run it as a regular user, you get a pop-up. You can also accept the license using the /accepteula parameter. I got the SYSTEM user to accept the eula by scheduling a task with the AT command:
at 05:00 /interactive /every:m,t,w,th,f,s,su "c:\yourdirectory\psexec.exe" /accepteula
After that, I went to the task scheduler, right-clicked the new task and selected run. PSEXEC can then be run by the system user.
(I deleted the temporary task after that)