Good Riddance to Bad Rubbish
If Charles and Marcee represent the kind of good side of John's delusions (hey, at least they're friendly ghosts), Parcher symbolizes the scary side. Instead of supporting John and offering him love and friendship, like Charles and Marcee do, Parcher represents the paranoid side of his hallucinations.
Unfortunately, John is trained to find and analyze patterns everywhere—it's why he's such a brilliant mathematician, after all—but that habit is part of his delusions, too. He thinks he sees secret codes everywhere, which is why Parcher (he believes) is using his talents to help out the U.S. in the Cold War.
And so, John believes that he's become involved in secret file drops and even trades gunfire with Russian operatives. Yeah, yeah, we know, it sounds far-fetched, but John definitely thinks it's happening.
Things eventually come to a head when, well, everyone else notices that John's become super paranoid…to the point where John has to be hospitalized.
John could function when just Marcee and Charles were hanging around, but Parcher and his secret missions really keep John from living his life and being able to work.
PARCHER: I told you attachments were dangerous. You chose to marry the girl. I did nothing to prevent it. The best way to ensure everybody's safety is for you to continue your work.
JOHN: Well, I'll just quit.
PARCHER: You won't.
JOHN: Why would I not?
PARCHER: Because I keep the Russians from knowing you work for us. You quit working for me, I quit working for you.
And so, as he did with Marcee and Charles, John has to give up engaging with Parcher.
It's easier for John to want to ignore Parcher, of course, since Parcher is kind of terrifying. However, because John's talents and work are related to recognizing and analyzing codes, the hallucinations featuring Parcher and his secret missions are pretty hard for John to resist.
John doesn't want to shake off his talents, after all, right? But that's the challenge: getting rid of or at least ignoring the hallucinations while still letting his "beautiful mind" still kick into overdrive with numbers, patterns, etc…just in a non-scary, non-delusional way this time.