This seems very strange. They were at the gate ready to board for BEY and some German officials were giving exit stamps, but refused to do so for one parent? This sounds like the German passport control employees thought that there was something wrong with the documents of the one parent.

However, I repeat my question: Parents are landside at FRA. After check in, they would go through exit passport control before boarding flights outside of the schengen area. Either the passport control officials stamp all (nonEU) passports or parent requests the stamp. This would seem to statisfy the requirement, no?

Alternatively, suppose one buys a fully refundable ticket to someplace outside of schengen/EU, checks in, and goes through passport control at FRA. That generates the exit stamp. Then they change mind, cancel flight, and ask to be escorted back through immigration to again go landside at FRA. They've been stamped out by then, so the requirement should be satisfied, although this has a bunch of problems, including the purchase of a fraudulent plane ticket as the passenger never intended to fly. Also, I'd be more anxious to try this on a mideastern passport than one from USA, Canada, Australia, Japan, etc. so I'm not recommending it, just pointing out that it should be easy to get an exit stamp at FRA somehow.

How about taking the train into Switzerland for a day? Does this generate the EU exit stamp? I suspect not as I rarely see immigration officials on European trains, but the situation might have changed in response to all the refugees in Europe.