Frequency compensation and feedback capacitance
An unstable closed loop system can be made stable in two ways: -


Lowering the open loop gain or,
Ensuring that the open loop phase change does not reach 180 degrees before the open loop gain drops below unity.


Clearly, for people who make opamps, they like to sell you something that has a massive DC open loop gain so, if the overall gain is dropped (to fix instabilities when running closed loop) then the device will look unattractive in applications that require ultimate DC precision.

That’s the back story for opamps but, it applies to any control loop; good positional accuracy is usually a strong desire whereas speed of response might be seen as secondary.

So, step in the miller capacitor. It is used extensively in opamps to ensure that the overall phase change from DC to high frequencies doesn’t reach 180 degrees by the time the open loop gain has fallen below unity.


  But sometimes I have read that these parasitic capacitances are
  terrible regarding stability


If you have a control system that is stable but has a phase margin of only a few degrees then adding any more phase angle without seriously reducing gain, will usually result in instability. However, if the phase change is added such that it dominates the existing poles of the system, then stability can be regained.

It is this last part that works in an opamp. Without the miller compensation, the opamp might undergo massive phase changes at really high frequencies (and well below the point where gain has dropped to unity) and, if high levels of negative feedback were applied it would become an oscillator.

So, the miller capacitor is used to dominate those areas and force the gain to fall below unity before the phase change can reach 180 degrees. But, if your miller capacitor circuit is only slightly significant at too high a frequency, it can cause instability to an otherwise stable system.