Such a troublesome little letter in Italian! It is required because "permettere" is one of those verbs that govern the use of "a" after them. "Permettere a qualcuno qualcosa" (To permit someone something) similar to "dare qualcosa a qualcuno" (To give something to someone.) For more information try this web-site: http://italian.about.com/od/verbs/a/aa031908a.htm
Yes, it's "permettere qualcosa a qualcuno" (to permit something to someone / to let someone do something); i.e. the direct object is what's being permitted and the one being allowed is the indirect object.
So earlier I had translated this same sentence to "Lei ha lasciato suo figlio correre" (and had it marked correct! ) - So I'm understanding from this discussion the "a" is necessary for the "permesso". But I keep posting similar questions (sorry) really trying to understand the use (or not!) of "a" or "di" with the infinitive. So why is the "Lei ha lasciato suo figlio correre" not requiring "a" or "di" in front of correre and yet somehow "Lei ha permesso a suo figlio di correre." requires "di" and why not "a"? My poor brain hurts, but thanks :)