UPDATE: My parents got a balcony room! Per the advice here on the forum, I checked the website several times a day. This morning I saw a balcony stateroom, so I immediately called my mom, who immediately called Cunard. They don't have a stateroom number assignment yet, but they did get a guarantee. They paid and all is well!
Thank you all so much for the wonderful advice! ☺️
They never received an email with an offer of availability. I called about a dining preference change a little later this morning, and the woman said they were "very lucky" to get a stateroom, and that she just got off the phone with someone who she put on the waitlist at #51.