I wonder if that protects the person who makes the parody, rather than the app developer who make $4K a month. Lawyers will not typically go after targets that don't have money for payouts.

Only if you are parodying the content used.

If you use copyrighted content to parody something else, you aren't protected. Penny Arcade had to pull a comic back in the day when they misunderstood this.

I'd say just be ready to take down an image if someone approaches you and you should be fine.