no such rule, it's a myth. But, if immigration thinks you are milking visa exempt stamps to live here, then prepare to be pulled aside. In the OP case they are a genuine tourist staying a few months at a time, no issues.
you can definitely apply for an extension in country, but it's a dog and pony show. Good luck. Don't leave it to the last minute in case things go south and you need to bounce.
The entry stamp on June 9th appears to have the wrong permission to stay date, it says it's a 60 day entry but the permission is only 7 days. I can't read the hand written notation under the stamp