Question: I was reading about diodes and I know that they have a certain parasitic capacitance.
Now, according to Wikipedia, 

> 
> At low frequencies parasitic capacitance can usually be ignored, but
>  in high frequency circuits it can be a major problem.
> 
> 
> 

I have trouble understanding what exactly happens, so if you could come with some explanations that'd be great. 

Is this because when it comes to capacitors, voltage lags behind, or is it something else?

Answer: 
Often you want to use a diode to block currents flowing in a certain direction. But the parasitic capacitor will pass high-frequency current signals, regardless of phase.

Often you want to use a diode to provide a high-impedance in normal operation (while providing a path to divert unwanted signals like ESD strikes). The parasitic capacitance causes the diode to pass high-frequency signals rather than block them.
At low frequencies capacitors appear as open circuits, which can be ignored. At higher frequencies their impedance starts to decrease, so they can't be ignored any more. This has nothing to do with the diode, it is common to all parasitic capacitors.