If the primary member is travelling with minors who hold status in their own right then, in practice, my understanding is that the minors would be granted lounge access in their own right, just as if they were entering the lounge by virtue of travelling in a premium cabin with an accompanying adult.
The minors would remain, at all times, the responsibility of the accompanying adult however.
Also in practice, they would be able to avail themselves of facilities such as priority boarding.
However, the letter of the law (as linked to above) is that the only benefit that the minor has rights to is earning and redemption of mileage. Technically speaking, there’s no provision in the Ts&Cs for the earning of tier points either, but in practice (again!) they do indeed earn them.
What they wouldn’t be able to do, though, in theory, practice or otherwise, is guest someone into the lounge themselves – say if the accompanying adult had no status.
Visions of Stewie from Family Guy rather come to mind when considering that particular scenario.
Note also that in the case of a minor travelling unaccompanied in a premium cabin, they would not be allowed access to the main lounges and would instead be looked after elsewhere by airline staff, including at T5 in special children’s lounges.
And with reference to the earning of tier points by the paying party, as opposed to the travelling party well, firstly, as mentioned above, the Ts&Cs don’t actually provide for TP earning by minors anyway and, secondly, it’s one of the principal tenets of the BA FFP (and most others) that the ‘traveller earns’.
Otherwise, we’d have CEOs, Managers, PAs and Receptionists who’ve never set foot on an airline brandishing the shiniest of cards, whilst the poor foot-soldiers received nothing for their discomforts.
Parents paying for tickets for their children is, practically-speaking, no different from companies paying for employees or indeed any other third party doing so for another, and the sheer scale of administration necessary to prove familial relationship, source of funds, correct re-posting of points and then continuous audit would be colossal.