I was researching a somewhat similar problem. The specifics of the problem was that every now and again our POP3 server became incredibly slow. At this point we noticed many connections by running this command:
root@cm:~# netstat -an|grep :110 |awk '/tcp/ {print $6}'|sort|uniq -c
33 CLOSE_WAIT
2 CLOSING
134 ESTABLISHED
1 FIN_WAIT1
2 LAST_ACK
1 LISTEN
13 SYN_RECV
19 TIME_WAIT
Notice 134 x Port 110 connections.
At this point when we TELNETed to the local server on port 110 we could see the problem, very slow response before you get the banner. In many cases our clients actually phoned to say they are having timeouts. We knew it wasn't the server because we had just recently upgraded the hardware.
We established Plesk uses Courier IMAP and POP3 daemon so that allowed us to do a more generic search of this problem. We also established when POP3 is slow IMAP is flying. First we were put off by this article which indicates due to strict compliance with RFCs Courier's POP3 daemon suffers from performance problems:
http://www.unwin.org/postfix/slow-pop3d.html
What this means is that pop3dserver must read the entire mailbox, character by character, making sure to count all the newline characters twice.
Luckily we also stumbled upon this article:
http://kb.mediatemple.net/questions/259/(dv)+HOWTO:+Raise+Courier-IMAP+Connections
By default, Plesk and the Courier-IMAP email server drastically limit the number of inbound connections to prevent users from opening up too many concurrent sessions. Unfortunately, this restriction can impact legitimate users who have multiple computers connecting to the Courier-IMAP server from behind a firewall or a single computer that runs an IMAP client that takes advantage of mailbox caching.
Plesk comes configured with a limit of 4 connections per IP address and a limit of 40 connections total. Modern IMAP clients such as Mozilla Thunderbird use mailbox caching to open up multiple connections to increase performance. In the case of Thunderbird, it opens up 5 connections by default which is already 1 connection more than Courier-IMAP's default restriction. Add another few computers behind a firewall and those additional users won't be able to connect at all since a single Thunderbird client is already utilizing all 4 connections.
We believe in our case the exact same is happening for POP3 connections as many of our clients come from a single IP and they have Microsoft Small Business Server with POP3 checking enabled.
We increased the limits as per the article, MAXDAEMONS was 40 and is 80 now, and MAXPERIP was 4 and is 40 now, and the problem seems to have disappeared. Parallels should really up the default limits on Plesk to avoid other people having the same problem.