No you cannot catch any kind of cancer, but BMAC is definitely correct, if you drink out of his glass, it is hard enough for your Dad to fight off bacteria, germs, infections etc, with cancer. You could make him sicker and more miserable than he already is, but he cannot make you sick by giving you cancer.
I am sorry that your Dad has cancer, hope all goes well. I know someone who had throat cancer, they had surgery, chemo, etc., and that was 10 years ago and he is still here. Good luck.