Final answer:
The palatoglossus muscle is innervated by the vagus nerve. It is involved in swallowing by elevating the back of the tongue and is part of the parasympathetic nervous system functions controlled by the vagus nerve.
Step-by-step explanation:
The palatoglossus muscle is innervated by the vagus nerve (cranial nerve X). This muscle plays a role in elevating the back of the tongue during swallowing. While the facial and glossopharyngeal nerves are also crucial in the oral cavity for functions such as initiating salivation and conveying gustatory stimulation, it is the vagus nerve that innervates the palatoglossus. The vagus nerve is responsible for several parasympathetic functions, including the control of muscles in the pharynx and larynx which are essential for swallowing and speaking.