What is the purpose of capacitors parallel to crystal?
The capacitors are effectively in series with each other and the combination is in parallel with the crystal. The total "load capacitance" is thus (ignoring the oscillator circuit and stray capacitances) 

\$C_P= \frac{C_1 C_2}{C_1 + C_2} \$ 

In a  typically microcontroller or similar circuit there is an amplifier input connected to one side and a current amplifier output connected to the other. Both have a bit of additional capacitance that adds to the load. It's called a series resonant circuit

The oscillator is called a Pierce Oscillator. 

There are two things that help you determine the value. First there may be suggested or limit on the maximum capacitance from the microcontroller or chip manufacturer. Best stay within that or your oscillator may not start reliably. Secondly, you would like, for maximum accuracy, to match the specified load capacitance on the crystal datasheet. That means taking into account the input, output and stray capacitances. The effective total load on the crystal is: 

 

So you want to pick C1 (usually = C2) such that CL is close to the number on the crystal datasheet. 

In general, it's not that fussy usually and you can use a few pF (like 5pF) for the values of Ci and Co and Cs. Because the crystal itself is a very high "Q" resonator, slight mismatches in the capacitance won't have much effect. If you're trying to keep time-of-day, you may wish to be more fussy and/or adjust the load capacitance, but usually 100ppm either way doesn't make so much difference. 

There are enough variables that you may wish to use whatever the recommended crystals are from the chip maker. In general there no guarantee that your oscillator will oscillate otherwise, since chip makers very seldom specify parameters such as transconductance  (gm) limits over temperature that could be used to guarantee operation. There is thus ample opportunity for the chip maker to blame you and the crystal maker, and the crystal maker to blame you and the chip maker (the uncomfortable common factor being you).