@Sako0 (by the way, I have no special knowledge or expertise on the topic of working with OSTR in San Francisco. I am just an SF host like you are.)
I would imagine the utility bill was mentioned, but not the most important thing. Just guessing, I would think the most important thing is the occupancy. The person getting the STR license is supposed to be the person who lives in the listing, so technically it is your parents who are supposed to get the license. Do they live there full time? (Or at least 275 days a year?)
Maybe they should apply. (Meaning you fill out the forms on their behalf; they should certainly read and sign them.)
As to your rating and such: if it comes down to having to cancel, do NOT do it yourself. Explain to Airbnb about the STR license and they should be able to take care of the cancellations for you, so that they are penalty-free. Airbnb is not interested in punishing you because the OSTR is being difficult. Airbnb is usually *very* interested in taking the OSTR situation into account. (Airbnb had to wrestle with the city for years about this, after all.)
If nothing else, this is certainly an extenuating circumstance for the host.