Thanks for the responses, everyone!
My only word of caution would be that all of the risks of AMS are still very real and very present. You should really bring your A game when you do this as the potential effects of the altitude and symptoms of AMS will not just creep up slowly. I have no idea what the ramifications are, but stay on your toes.
And please post a trip report. Good luck!
Neither my parents nor I have ever ridden any type of bike (mountain bike, road bike, dirt bike, etc.) above 11,000', so this will definitely be a new experience. We're planning on staying two nights in Mammoth to acclimate. I don't know whether the ride will happen this year, but if not, we'll definitely do it next year and will post a trip report afterwards.
I cannot imagine a couple of people in their seventies who aren't in very good shape managing to climb or descend that White Mountain with a mountain bike. They might be able to do the first 5 miles where the road isn't as bad, but that final climb....
I guess we have two choices when we get off the bikes at mile 5. Option 1 would be to lock the wheels of the bikes to each other, leave them a few feet off the trail, and unlock them on the way back down. Option 2 would be to attach our daypacks to the bikes and walk/push them to the summit. I'll probably go with option 2 if they're still feeling good at that point.