Zener diode parallel to RC circuit
That's not a zener. That is a Schottky diode needed for its fast response on the secondary side of the flyback. The RC in parallel will be acting as a snubber to manage its response (power loss, ringing)

The primary side D+RC is to deal with the leakage inductance of the transformer. The diode ensures the RC is in circuit only during turn-off. The C will absorb the energy stored in the leakage inductance (increasing the voltage to a controlled level) and the R is to discharge before the next switch event