bmwe92fan is right. Here is
the rule on United's website.
United Global Services® members departing Frankfurt, Germany or Munich, Germany and traveling in United Global First may access the Lufthansa First Class Lounge. Some restrictions apply.
No idea what the restrictions are. Since LH's own lounge rules allow a passenger with access to the FCL to invite a guest with a BP for the same flight I expect there's no problem taking the 13 year-old son with them to the FCL.
The only feasible FCL on the return journey will be the FCL A, which in a way is nicer than the FCT with a good tarmac view. Unfortunately it's a Schengen lounge which is convenient when they arrive from ATH but the OP and the family will have to leave the lounge to go through passport control to Z departures in good time.