swelling around the facial shouldn't cause any pain...if anything, it'll trigger a full, ipsilateral face droop (CN VII palsy).
i could imagine the parotid gland would swell for multiple reasons, but the most common is probably due to inflammation. inflammatory molecules released from vessels and molecules released from startled mast cells around the parotid gland should trigger nociceptors (whose cell bodies are located in the trigeminal ganglion) to send an AP to the spinal nucleus of CN V. secondary neurons from the spinal nucleus of CN V desiccate around the caudal medulla and carry an AP along the contralateral trigeminothalamic tract somatosensory cortex/insular cortex (first stopping at the VPM nucleus of the thalamus).
all sensation in the face comes from CN V (with some sensation of the ear coming from CN VII, CN IX, and CN X).
NOTE: trigeminal GANGLION...not nucleus.