Suddenly and without changing anything in the zp-core OR theme files, I get this message when I try to upload images:

Warning: move_uploaded_file(/srv/jackdaw/www/www.spoenk.nl/zenphoto/albums/Fotografie/Dieren/Varkens/boom3.jpg) [function.move-uploaded-file]: failed to open stream: Permission denied in /srv/jackdaw/www/www.spoenk.nl/zenphoto/zp-core/admin.php on line 371

Warning: move_uploaded_file() [function.move-uploaded-file]: Unable to move '/tmp/upload/php2vJuFE' to '/srv/jackdaw/www/www.spoenk.nl/zenphoto/albums/Fotografie/Dieren/Varkens/boom3.jpg' in /srv/jackdaw/www/www.spoenk.nl/zenphoto/zp-core/admin.php on line 371

Warning: Cannot modify header information - headers already sent by (output started at /srv/jackdaw/www/www.spoenk.nl/zenphoto/zp-core/admin.php:371) in /srv/jackdaw/www/www.spoenk.nl/zenphoto/zp-core/admin.php on line 398

Uploading new images by ftp works fine.
I haven't got a clue how this could have happened. I refreshed the database (though not certain this would effect the problem), emptied the cache, logged in and out, restarted the browser etc...

Maybe you accidentally changed the permissions by using ftp, not exactly the error but take a look here:
http://www.zenphoto.org/support/topic.php?id=904&replies=29#post-12267