Parallel PoE injector from pc psu?
PoE doesn't have "power pairs" and "data pairs". All pairs are both power and data. And those 4 pairs necessarily connect to the non-PoE switch as well. It should tolerate the excess voltage, but it's not designed for that. 

As Jasen notes, standard PoE is 48V, not 12V.

The camera will indeed negotiate PoE power.