Does liver disease cause parotid swelling?

Usually not. Generally no, but there are many things which can cause swelling of the parotid glands (spit glands in the cheeks). You should have it checked out by your md, but in general, painful swelling of the parotid glands can be initially treated with hot compresses, good hydration (lots of water) and sour foods/liquids taken 5-10x/day (vinegar, lemon juice, sour candies, etc).