I’d always put myself out for a colleague facing a one-off emergency or similar, but the problem for op’s colleague is that this is an ongoing situation.
This, in a nutshell, is why as a senior manager I generally speaking refuse to give any special "parental arrangements" that everyone else doesn't get at comparable terms ...
... and it is EXACTLY so that I can then actually step in and pull the "but this one is different" card when something like the OP's situation arises, i.e. when someone's in a real pickle. And my teams will virtually always agree because for them, this is not an ongoing situation but the exception!
I fully agree, by the way, that this sounds like a management problem on multiple fronts. That said, I do have different kinds of employees (not in the care industry, but some of the business I manage does include critical infrastructure stuff that requires holiday cover), and I don't generally think that letting them come to their own agreement is always bad management. Some people prefer it and will be less resentful if they get the chance to negotiate with each other. Others, however, due to a variety of factors, I would not dream about suggesting this to.