Let’s have a look at the law. I imagine the police are thinking of Section 5 of the Public Order Act. It says this (and I’ve left out bits that do not seem relevant from what you’ve told us):
Harassment, alarm or distress.
(1) A person is guilty of an offence if he—
(a) uses threatening or abusive words or behaviour, or disorderly behaviour, or
(b) [not relevant]
within the hearing or sight of a person likely to be caused harassment, alarm or distress thereby.
(2) [Not relevant]
(3 )It is a defence for the accused to prove—
(a) That he had no reason to believe that there was any person within hearing or sight who was likely to be caused harassment, alarm or distress, or
(b) [Not relevant]
(c) That his conduct was reasonable.
So, first of all were his words “threatening or abusive”. You told us he was asked and repeated:
“have you seen the Allah Akbar Drop the bomb sponge bob square pants vine'
Threatening? I think possibly not. If so, who was thretened? Abusive? Abusive towards whom? What was the nature of the abuse?
Then:
(3 )It is a defence for the accused to prove—
(a) That he had no reason to believe that there was any person within hearing or sight who was likely to be caused harassment, alarm or distress,
“My son had no idea the lady was in the park, he explained to the police office he didn't know hat it meant and the police officer had to tell him.”
I think you need to establish precisely what is alleged. If the allegation is as straightforward as you have explained to us then I think it fails under para 1(a) in that it could not be construed as threatening or abusive. Even if the police are not with you on that then the statutory defence (para 3a) kicks in. Note that for such a defence to succeed you only have to show that your son had "...no reason to believe that there was any person within hearing or sight who was likely to be caused harassment, alarm or distress," and NOT that there actually was not any such person.
A trip to the Citizens Advice Bureau may point you towards getting a free legal opinion. If that opinion concurs with mine (which is that I believe the police are trying it on for reasons that escape me at the moment) I would tell the police to place their caution where the sun does not shine. They can then decide whether to take your son before the Youth Court where a more reasoned decision might be reached.