Q:

What does a capacitor parallel to a resistor do in the feedback of op-amp circuits?

What is 10Mohm for?
What would happen without it?

A:

The circuit is an integrator and in the absence of an input voltage, the op-amp output should stay held at the voltage it acquired prior to the input becoming zero. However, nothing is perfect and the op-amps input imperfections will tend to push or pull the output towards either power rail (but slowly). To this end, the 10 Mohm resistor will try and counter that by discharging the capacitor slowly. Now the op-amp output will gradually drift back to centre-rail. 

