The only thing that immediately springs to mind is that the parent might not realise that the alleged behaviour amounted to a crime.
That's also my best guess.
You can see why - the parent may think he's old enough to hop into the sack with someone also aged 16 or more, so sending a few naughty pics could be seen as less serious, therefore not an offence.
The parent is wrong about that, sending naughty pics is an offence for a 16 or 17-year-old, even if the teenager is keen to do it.
But the parent still believed the only redress would be via the BBC.
They dragged their feet, so enter the Currant Bun which did its usually excellent job of publicising the story, thereby concentrating the minds of the employer.
Naming the presenter at this time is an ethical and legal can of worms.
The case of Cliff Richard, who got a nice few quid after he was effectively named as a suspect, has at the very least pushed back the name/not to name pendulum.
There remains a public interest justification, but one might think this one falls into the interesting to the public rather than in the public interest category.
Even that's not straightforward, say the presenter worked in children's television or occasionally covered events where members of the public of all ages were present.
It would then be arguably in the public interest to name to enable parents to prevent their children coming into contact with him.
Another thing that has happened, and may yet happen in this case, is that foreign media publish the name, believing they are outside the reach of British regulators and courts.
A quick google suggests this story has gone around the world, suggesting the interest is there.
There is then a 'the cat's out of the bag' argument for British media to publish the name.