Can my parents extend their stay in Thailand if they experience an overstay on their 30-day visa exemption?

As some people mentioned, when you leave Thailand by AIR with one day overstay, the fine would waived, but you will be stamped mentioning overstaying. And IF you get into the police before you reach the airport, you will be in trouble. The possibility would be low but not zero. There is downside of this overstay stamp, such as you cannot apply for TR in some consulate around SEA or Elite program.

We do advice NOT to plan to overstay intentionally. But it's passport holder's decision after all. They still have time to decide if you want to have overstay in their passports or try to avoid.

Yes best option is apply online for the single entry tourist visa (60 days). Or they could do a trip to VN, Laos, KL, while here to sight see and reset their 30day.

What time is their flight on 4th October?

If they're internet savvy they could apply for a tourist e-visa before traveling to Thailand and then they won't need to worry about getting an extension while they're here.

UK passport holders will be stamped for 30 days on visa-exempt entry. You can apply for a 30 days tourism extension for a 1900 thb fee at the local immigration office. There is no 1-week extension. But every extension is the same fee regardless of the period, 1 week, 1 month, or 1 year.

They still have time, maybe consider getting a Single Entry Tourist Visa for each. The official fee is cheaper than the extension fee inside Thailand, they don't have to spend hours or 1 day to go to the immigration office during that 31 days. Also, less hassle with airlines to check in without a proper visa or return ticket within 30 days of stay that visa-exempt allows.

BTW, 45 days on visa-exempt program had finished on March.