First thing I would check is if SELinux is causing this issue. You may be able to do this by temporarily disabling it, or by looking at /var/log/audit.log.
If thats not the case, you need to establish where the files are being written to - It may well be that its where you expect, but as the errors are showing relative addressing the problem could be elsewhere. (I don't use PHPBB, but I guess if you look for "Found, Unwritable" in the PHP code, you can hack a line near it to show the full path, maybe with something like echo $dirname($path variable). Alternatively there may be a config file you need to set which has the base directory for the path.
Also, as you have "chmod" access, try chown those directories to the web user (normally http, httpd or www-data - you can look which one by looking at who is running the web server or the usernames in /etc/passwd. I don't think running chmod 777 is a great idea although it will allow files to be written regardless of owner - while running chmod 002 is clearly wrong, allowing no one to read files, and only non-owner and non-group users to write files )
You should also check that all the parent paths are readable by the web user, and make sure the directories you are looking for exist.