Answer:
The 3rd option, The participial phrase "barking in the yard" modifies "dog".
Explanation:
A participal phrase consists in a participle (past participle or present participle <verb+ing>) and a complement to the participle (nouns, pronouns, direct or indirect object, etc.). These phrases fuction as adjectives, it means, are intended to modify a noun or pronoun in the sentence.
In the sentence, "The dog barking in the yard for the last three hours is giving me a headache."
Barking is the participle (verb bark + ing)
in the yard for the last three hours is the complement of the participal phrase
The phrase is explaining why the dog is giving me a headache (because the dog has been barking in the yard for the last three hours), so is directly connected to the noun "The dog" modifying it.
