I just upgraded to 1.5.8 (following the instructions). The setup pointed me to set permissions which I clicked and now I only get a white / blank screen.

PHP Version is 7.4.16

I double-checked the permissions on files and folders and they are according to the recommendations in the documentation.

In the error_log I find the following entry:
[02-Jun-2021 17:45:44 UTC] PHP Parse error: syntax error, unexpected '=>' (T_DOUBLE_ARROW) in index.php(11) : eval()'d code on line 86

Any ideas?

The permissions have been set too strict probably .Often the setting is fine for the server just setup reports (or recommends to review) even stricter ones, But often these break things. It's a bit of trial and error what works on a server. Having the standard 0644 is mostly fine.

Please try to reset the permissions via FTP and see if it then works.

Actually, this sounds more like a corrupted config file. How does line 86 of your zenphoto,config.php read? Maybe show 5 or 10 lines on either side as well.