From your mention of your "old server" and "new server", it sounds like you performed what Plesk calls a "migration" to a new server with a newer version fo Plesk, noit what Plesk calls an "upgrade" (in-place upgrade of Plesk on the same server).
I had *many* problems when I migrated a VPS that included active Mailman lists from a virtual server with an older Plesk and OS version to a virtual server with Plesk 12.5 and a newer OS:
https://hasbrouck.org/blog/archives/002074.html
You should be aware that (a) the Plesk migration tool does *not* migrate any Mailman lists data or configurations, (b) if you are not careful to shut off the mailing list service on the new server before migrating and completing configuration of Mailman lists, you might spam all the members of all lists!, and (c) the version of Mailman available for installation through Plesk may be (and probably will be) a substantially older version than the current version. If you want a recent version of Mailman, you may have to compile, install, and configure it manually.
If an older version of Mailman is sufficient for your needs, here's what worked for me to migrate Mailman lists (settings, configuration, boilerplate, list memberships, and per-member settings) from one CentOS Plesk VPS to another with a different version of CentOS and Plesk:
(1) If any Mailman lists are in use on the server, turn off "welcome" messages on the new server (server-wide setting) before migration:
http://kb.parallels.com/en/113641
If you don't do this, you will probably spam all members of all lists with spurious "welcome" messages!
(2) Stop mail services on the new server.
(3) Make sure Mailman is installed as a Plesk option on the new server. Depending on the Plesk and OS version, it may or may not be installed by default. If it sisn;t installed, it should be possible to install it through the Plesk installer GUI.
(4) Migrate Mailman settings for lists and subscribers by manually copying the entire contents of:
/var/lib/mailman/lists/
from the old to the new server, preserving permissions.
Make sure to do this *before* restarting mail services on the new server after installing Mailman, or data may leak through spurious messages or list administrivia messages broadcast to all subscribers instead of sent only to the list owner.
Note that the migration -- even if the Mailman versions are the same -- will initially reset all lists to Mailman defaults, including that they will be unmoderated discussion (not announcement) lists, and all subscribers' statuses will be reset to unmoderated. This is a potential security and privacy vulnerability, and can lead to administrivia (such as complaints about the spurious "welcome" messages) being broadcast as spam to all list members. If you choose not to copy over the full Mailman settings folder, you should at least check the announcement/discussion and moderation settings for all lists *before* starting or restarting mail services on the new server.
(5) Restart mail services on new server.
(6) Turn Mailman list "welcome" messages back on, per list, if desired.
Hope this helps. I can't promise that this will work for you, but it's what I eventually figured out I should have done. Good luck!