Question

we were impressed by their cooking.
subject
direct object
object of
preposition

Explanation:

To determine the role of "their cooking", analyze the sentence structure. The preposition is "by", and the noun phrase following "by" is the object of the preposition. "We" is the subject, and there's no direct object here (the verb "were impressed" is passive, with the agent introduced by "by"). So "their cooking" is the object of the preposition.

Answer:

object of preposition