What could cause swollen parotid glands?

Many things: Viral infection, bacterial infection, tumors from which pleomorphic adenoma would be the first.

Blockage: While my colleague is correct, the reasons given are rather more rare than the simplest explanation: blockage. Calcium deposits or firm stones of calcified elements can cause the parotid gland to swell. It happens when people even look at a pickle, an start the juices flowing. Sialography or a sialogram can tell where the blockage is and how best to treat it. Rule out significant disease first.