Any thoughts on what sort of cough this is or what it could be?

By the duration of cough it must be chronic bronchitis due to excessive mucus secretion in the bronchioles. Smoking actually aggravates it. A course of cefadroxil 500mg for 7 days- three times a day must bring it down. Bronchodilaters like salmetrol may be used if any obstruction is felt. Smoking 30 per day gives a count around 1.5 pack years medically, which is a 10 fold increase in risk for cancer. Get a hold of your dad and ask him to decrease and gradually cessate. Or else....!!