I finally reached someone at the IRS who answered the questions. It'll be a couple of months before I know if they were correct, but in case anyone else has the same issues, here's what I found:
First of all, reaching someone on the phone was much harder than it should be - not because of hold times (they have a call-back feature so you don't have to stay on the phone), but because it's not clear how to navigate the voice prompts. I tried saying "question about form 56", but that sent me to the form-ordering department, where the worker's only advice was "next time, don't say a form number or else you'll be transferred here."
I tried other responses, but none were recognized. Eventually, on another call a week later, I got so frustrated I accidentally said "question about form 56" again. But this time, the person in the forms department was experienced enough to know the secret. "Say "account-related question", she said, "and that should get you to someone who can answer your questions." She was right!
Since I live far from my parents, and getting their signatures for a change-of-address form would be all but impossible, the IRS rep said to do this:
1. File a form 56 for each parent. In about 30 days, I'll receive confirmation letters.
2. Then I can sign a change-of-address form (8822) as their representative, and use my mailing address.
3. I will be able to create online IRS accounts for each parent. Since my name will be on their accounts, I can verify my identity while setting up the accounts. (I confess to being a little skeptical about this. I really hope the IRS rep was correct.)
I could have put my mailing address on their tax when doing it in TurboTax, as @guywong said. A cousin told me he's been doing this for his parents, and it hasn't been a problem.
Fingers crossed all of this works.
