Can anyone explain how any such check might be enforced, and where ?
With OLCI you don't actually see anybody, and security are not looking for anything other than on-the-spot prohibited items. Likewise the gate check only does boarding passes and passport validity.
My feeling is that any regulations about "proof of parentage" are only ones that could be activated in a particular situation which has already come to the attention of the authorities, and they are watching out for somebody. There appears no standard check at all and I would guess the majority of those travelling with small children have no such evidence on them anyway.
In the US in particular there is a whole raft of publicity about carrying "certified documents" for such situations, the publicity appears to be a marketing effort by the notaries who do the certifying, and charge significant fees for their completely unwanted services.