The question is intrinsically acceding to the deceit of the system: it is questioning one specific "miracle" rather than the concept of "miracle" itself. If you view the broader concept itself you better understand where miracles come form. Because there's a fairly easy explanation for pretty much any (medical) miracle: survivorship bias and cherry picking.
You take one "one-in-a-million chance" and then run it several million times, and the odds become very good that you'll get at least one success. Hooray, miracle! Pay no attention to the millions of failures behind the curtain.
For example, let's run some Parkinson's statistics. They estimate that there are approximately 10 million people with Parkinson's worldwide. Let's suppose the disease is independent of religion, or at least the Catholic religion. Since approximately 16% of the world's population is Catholic, that gives approximately 1.6 million Catholics afflicted with Parkinson's. If it were a "one-in-a-million chance" (taken quite literally) to be cured in some sense during a given span of time (including the possibility that you never really had it, but had something else, as well as some intervening phenomenon resulting in a temporary suppression of symptoms, etc.), we'd expect there to be about 1.6 Catholics in the world to experience a sudden cure during that time span.
Now we could start throwing in other factors: how many of those had actually met/contacted/touched the Pope or been prayed for by him, how many of those were actively praying to be cured, how many of those were thinking of the Pope in particular, how many would convincingly claim they were, etc. It becomes increasingly hard to estimate all of the variables, but there's one inescapable fact: this was for just one disease! There are many horrible (and incurable) diseases in the world. The NHC estimates some 40% of Americans alone have a chronic (and essentially incurable) disease. When you actually start looking at all of the possibilities, you realize it becomes rather more miraculous for no one to have had a "miracle" cure from anything at all. It becomes rather expected that one or two of the survivors will point to the same globally prominent thing to explain their good fortune.
So your explanation in this case is that by some chance event (or calculated deception) some Catholic had something wrong with them, but then didn't, and happened to believe that the head of their religion was involved. I have no idea what that chance event was (nor do I have any evidence of a deception, to be clear), but this isn't necessary to understand that a success that exists essentially by the law of large numbers was singled out, and a massive pile of failures was ignored.