You will hear stories about how people phoned up their firm choice and persuaded them to take them after they were rejected. However, I suspect that is very rare once the decision is taken. It may be more common if UCAS Track is still showing you to have a conditional offer, because that means for whatever reason a decision has not yet been taken and a phone call may help.
What you have to appreciate is that University Admissions departments are a frenetic hive of activity in the days before you get the results. First, we have to manually confirm everyone who has made the grades. Then we have to see how many spare places we have. Then we have to decide, assuming we have spare places, which near miss candidates get those places, and we do that by looking at all the applications of near miss candidates. All of that is done through meetings and discussions and management diktats, all of which focus on a combination of student numbers and acceptable grades.
So if an applicant logs onto Track and sees that they do not have a place, one of three things is going on. It might be that enough people got the grades to fill the course places and the university cannot take any more students on that course (student numbers); it might be that their grades are too low to be considered for the course (acceptable grades), or it might be that their grades, whilst just off the offer, have been considered, but overall their application wasn't as strong as others in the same position (student numbers). None of those things can be changed by a phone call. In particular, whilst we have more administrative flexibility than we used to have in terms of student numbers, we still have to bear in mind class sizes, personal tutor allocations, accommodation spaces etc - we can't take everyone we want. And the overarching obligation of the Admissions Tutor in particular is to be fair to all applicants - why should one applicant who rings up get a benefit over one who may not, due to personality or background or whatever else. If I give one phoning applicant a place, there may be others who are at least as good on paper - should I not give them all places as well. So it's not 'just one place' - it may, to be fair, be many more. So any applicant needs to think 'why am I special?'; really, really wanting to do the course doesn't, unfortunately, make you special. The fair thing to do is focus on the application, rather than wait for phone calls.
One thing that does sometimes happen in a phone call is that mitigating circumstances are revealed which were not previously known about. This is always frustrating, because if they are known about before, they can be taken into consideration. I have been known to reconsider a decision if I'm told about important mitigating circumstances which have had a clear effect on results - but I can't always, as it depends on my numbers (if I'm already over target then I just won't have space), so I may also promise to make an unconditional offer if an application is made for the next year. So it is always important to inform Admissions of any potential mitigating circumstances.
In short, if you've got nothing new to add to your application, a phone call is rarely going to make a difference and it's probably best, from a parents perspective, to encourage your DC to accept the situation and move on. If you have mitigating circumstances, tell the university you've applied to sooner rather than later.
It's also worth saying that the fact that a course is in Clearing is not helpful if you've been rejected. Universities would always prefer people who make them their firm choice over clearing candidates, so if a firm choice candidate is rejected and the course is in clearing, they are looking for applicants with higher grades. It's important, I think, to understand that if one is rejected it will be for a good reason and the application will have been looked at.