When a pair of parallel lines is cut by a transversal, several angles are formed. In this case, angle \( p \) is an exterior angle made on the right by the intersection of the upper parallel line and the transversal, while angle \( q \) is an exterior angle made on the left by the intersection of the lower parallel line and the transversal.
Since angles \( p \) and \( q \) are both exterior angles created by the intersection of a transversal with parallel lines, they are actually corresponding angles. Corresponding angles formed by a transversal cutting through parallel lines are equal.
Therefore, the relationship between angles \( p \) and \( q \) is: \[ p = q \]
Thus, the correct choice is: \[ p = q \]