The parts of speech for the words in the sentence are:
Jim: noun. A proper noun
who: pronoun. A relative pronoun, refering to the subject Jim and Tiffany.
favorite: adjective. A word to describe cousins in this sentence.
just - adverb. Adverb of time - short time before now.
cutest - adjective - to describe dog.
running - verb - word that expresses action or state.
their - adjective. Possessive adjective (possessive dederminer)
behind - preposition (of place), showing the position of the dog.
the - article - definite article, refering to a car that is already known.