At least in downtown, "public parking" is rented by the city at the rate they set. You may claim the rate city sets is too low, yes. But that doesn't actually entitle you (or anyone else) to lay claim to parking spaces and auction them as you please - since, well, they don't actually belong to you.
This doesn't really have much to do with "publicness" actually. Rental car companies would be equally upset if people developed an app for car renters to extract a surcharge for handing a rental off to their "friends".
The poor can't afford smartphones. If your plan were implemented, how would they find parking? If they aren't able to participate in your system, then that will add a lot of time to their daily commute.
Poor people are already penalized for being poor due to overdraft fees, predatory credit, etc. Taking an extra hour of their life each day just so the wealthy can find parking more easily doesn't seem best.
But on-street parking in crowded urban areas with good public transportation? Absolutely not. Why should I take the subway or bus, yet have my tax dollars subsidize a select few who want to park on the street for free, or park for less than a market-clearing rate?
This app is great because it highlights how much money the city, and therefore taxpayers, are losing. I agree the app should be shut down, but only because the city should build its own so it can take the money directly.