Final answer:
The facial nerve innervates the skin around the ear and over the parotid gland, which is adjacent to the masseter muscle; the vagus nerve is not involved with this innervation.
Step-by-step explanation:
The nerve that innervates the skin on both sides of the ear and the skin over the parotid gland is the facial nerve (Cranial nerve VII). The facial nerve also controls the muscles of facial expression, as observed during various facial movements such as raising eyebrows, smiling, and frowning. Additionally, the landmark for the parotid gland used in clinical examination rolls over the masseter muscle, which is innervated by the trigeminal nerve (Cranial nerve V). The nerve to rule out in this context, given it has a different function including motor control of the gag reflex and regulation of visceral functions such as heartbeat and digestion, would be the vagus nerve (Cranial nerve X).