You can file a Form 8822, change of address, with the IRS and have both of your parents sign and date it.
I would have used your address on your parent's tax return instead of their nursing home when preparing their tax return. To my knowledge, there is no requirement to use your parent's physical address on the tax return. I've prepared many tax returns using a loved one's address on the f1040.
About Form 8822, Change of Address | Internal Revenue Service
Perhaps you can set up an IRS account the next time you visit your parents and use your address.
After the address has been changed, your parents may receive a notice from the State taxing agency where you reside, inquiring or demanding a state tax return. Just write back to the State and explain to them that your elderly parents are not residents of your State and they simply use your address for mailing purposes.
Alternatively, if you had signed your name as the preparer, you could file a Form 2848, Power of Attorney, have your Dad appoint you as an "attorney-in-fact", mark the box that reads, "Check if to be sent copies of notices and communications". IRS will only allow you to represent your Dad for 3 future years, but you can put in all the past open years. Assuming you're not an attorney, a CPA or an enrolled agent, you will be considered an "unenrolled agent and you will need a PTIN first. So this is a lot more complicated.
About Form 2848, Power of Attorney and Declaration of Representative | Internal Revenue Service
