Yes planes can fly in heavy rain.
As long as the rain does not get too bad and as long as the rain has no severe thunderstorms or tornado etc with it the plane can safely fly in the rain.
Although plane pilots prefer it to not be raining when they fly they will still fly in the rain when it's safe to do so.
However if the weather turns really severe and theirs hail, high winds etc in the forecast then the airline will usually cancel the flight until the weather gets better.
I've flown in airplanes during really heavy rain and the flight went just fine.
It had me worried though but we were safe.