Puzzled with parallel forward biased diodes with common anode and different cathode voltages
if only the diode with -12V cathode voltage existed V6 would be -11.4V.


And if \$V_6\$ is at -11.4, then what operating regime is the other diode in?


  But maybe it will be how can we know?


The general approach to these problems is, 


Guess which diodes are forward biased and which are reverse biased.
Check whether your guess leads to any logical contradiction or non-physical result.
If it does, go back to 1. and make a different guess.


It's possible (in a more complicated circuit) that more than one guess will produce no contradiction. In that case, the circuit may have multiple stable states, and the actual state will depend on the history of the input voltages over time, or may even develop randomly (through metastability, for example).