How are you parting the channel? Simply closing the channel window won't work unless you have a script specifically set to do so. If you have one, then it could be this that is causing the issue. /Part #channel part-msg-here
should work just fine.
It may not show up on some channels if they have certain modes set. For example, the +m mode will suppress /part messages and often /quit messages too. If the channel is set +m then it could be that. If the part message has colour in it, and the channel has the +c mode set on it, that could also be the issue. Check out the network's #Help channel for any other modes which could be causing this to occur.
To ensure a script isn't causing the problem, use /!part #channel msg
, or use /!remote off
(or indeed use both, but not necessary!)
The mIRC client itself has nothing to do with part messages so it's not a bug in mIRC or anything - although unlikely, it may even be a problem with the IRCd.
Best of luck