It sounds like your dad has caught bronchitis. This is an upper respiratory tract infection like the common cold, but involves the lower airways as well. The symptoms are similar to the cold, but the cough associated with bronchitis can persist for a month after the other cold symptoms have gone away. The reason the Mucinex isn't working is because Mucinex does not really work that well. This is especially true for this type of cough. Mucus secretion is not the only reason he is coughing. The cough is likely due to inflammation in his airways and nasal drip coming from his nose. To combat this, he needs medicines which will calm down this inflammation and dry up the nose.
I suggest that you schedule an appointment for your dad to see his primary care physician
. He or she can take a detailed history of his symptoms and perform a thorough lung exam. If his doctor
thinks this is a post-viral cough, then he will benefit from a short course of a steroid inhaler and perhaps some decongestants. If his doctor is worried that the cough represents a bacterial infection such as a pneumonia
, then he may get a prescription for antibiotics. I hope he feels better soon.