Dad has MRSA in his lungs?

So sorry that you are having to deal with MRSA on top of caring for your dad on hospice.

MRSA needs an access point past your body's normal defenses, which is why good hygeine practices and envioronmental cleaning are effective. About 33 % of the population actually carries staph in their noses. That said, I wouldn't want anyone coughing into my face, because getting droplets in my eyes makes me feel gross. If it's giving you the heebie-jeebies, throw on a pair of glasses and/or a mask while doing care.
The second article below seems to indicate that if you get a small bedside HEPA filter or essential oil diffuser it might help, but not sure how legit the site is.
https://www.osha.gov/SLTC/etools/hospital/hazards/mro/mrsa/mrsa_employer.html
Airborne MRSA:
https://www.staph-infection-resources.com/prevention/airborne/

Google: MRSA contagious. There are articles from CDC, Mayo Clinic regarding MRSA. As Elizabeth Y mentions, besides gloves, wear a face mask ( same type you wear against Covid) when you enter his room and tend to him.