Ive installed it before and it will work on plesk :-)
You want to create your mysql database first before you do anything. Go into plesk control panel and click on the domain you want to use the forum on. Then hit the database button , it will say add new database you add in phpBB2 hit next, it will ask you for a username use anythinguwant and hit next, it will ask for a password use anythinguwant2
That is your database phpBB2 added in.
Now upload your entire phpBB2 folder into httpdocs.
Change the permissions on config.php to be writeable by all.
Now give
http://www.yourdomain.com/phpBB2/install.php a blast and it should bring up a setup screen. Blast in your details here... some of which are going to be the username and password you set for the phpBB2 database in plesk,db setting=localhost,one setting is mysql 3.2x or something,i think it asks for the path to your main files and thats going to be /usr/local/psa/home/vhosts/domain.com , does it ask for a mail setting hmmmmm if its does its /usr/sbin/sendmail
Anyway you will figure all that out then just hit the submit or whatever there is at the bottom :-) Jobs a good un!!
Thats you going now just blast back to config.php in httpdocs again and change the permissions to be writeable only by yourself !
Now remove install.php upgrade.php and update_to_201.php files
Your forum admin is going to be at ...
http://www.yourdomain.com/phpBB2/admin/index.php
and the forum itself will be at..
http://www.yourdomain.com/phpBB2/index.php
This isnt meant to be 100% accurate its just to give you an idea

I am 100% confident you will get it working with this post! :-)