What's more natural for a mother to say to her kid - when he wants to do something inappropriate, out of the following 2 choices?
"I don't let you do that" Or "I don't allow you to do that"?
Or they are both equivalents in this context?
Neither.
The form is "I won't let you do that" or "I won't allow you to do that."
The first is more likely, unless the mother is deliberately exposing her offspring to wider usage of language in order to expand his or her vocabulary.
Also possible is:
There are other forms.
But you won't ever find "I don't let you do that" used.
On the other hand, if you were talking to a third party, you may say:
EDIT: As pointed out by the-baby-is-you: the most idiomatically natural way to say what OP is trying to communicate is in fact, probably: