Final answer:
The Eustachian tubes are responsible for equalizing the pressure within the middle ear while swallowing, connecting the nasopharynx to the middle ear.
Step-by-step explanation:
The part of the upper respiratory system involved in equalizing the pressure within the middle ear while swallowing is the Eustachian tubes. These tubes connect the nasopharynx to the middle ear. The primary function of the Eustachian tubes is to allow pressure to equalize across the tympanic membrane, helping to prevent pain and discomfort in the ears, for example, during changes in altitude or when swallowing. When the Eustachian tubes open during activities such as swallowing, they balance the air pressure on both sides of the eardrum, which is essential for proper ear function.