Well, podPress 8.8.8.x has a new section at the Feed/iTunes Settings page. It is called podPress Feeds. This section contains besides other options the possibility to activate or deactivate these Feeds.
If you have updated from 188.8.131.52 to 184.108.40.206 directly, the first Feed should have been a Feed with the slug name "podcast". All the settings for this Feed are the settings for exactly this http://dysfunctionalparrot.com/?feed=podcast Feed.
So, one thing you should not do is to uncheck the Active box of this Feed with the slug name podcast. That would switch off this /?feed=podcast Feed.
If you change a slug name, activate or deactivate such a Feed then you should save your Permalink settings. Saving the Permalink settings may also help after the activation of the new podPress version.
That are the two most important things you should be aware of.
Please, try it again or describe in more details what you have done after the upgrade attempt.
Interesting for me would also be: Have you upgraded from 220.127.116.11 to 18.104.22.168 directly? Were the settings (like description, summary, owner email etc.) of the "podcast" Feed the same as the settings for the main blog Feeds?