The impedance (and capacitance) of a diode changes very abruptly when it stops conducting (after reverse recovery is completed). This together with a series inductive component usually results in moderate voltage spikes or ringing. A parallel impedance (capacitor) helps to make this change not so abrupt and the spikes smaller.
The ideal solution is not a simple capacitor in parallel with each diode, though. An additional RC may be required to damp ringing. Also, a single C (plus optional RC) in parallel with each transformer secondary should be enough instead of four.