Translation:I do not let your mother let you drink beer.
Because the purpose of duolingo is not to give you phrases to memorize, it is to teach you how to use the language from the ground up. Language is inherently infinitely generative and you cannot speak a language unless you have the capacity of infinite generativity in that language.
I am not giving permission for your mother to give you permission to drink beer:( It sounds just like Peter Piper picked a pickled pepper. But the pepper Peter Piper picked was bitter. So Peter Piper picked another pickled pepper. No wonder I am not giving your mother permission to allow you to drink beer:D:):)
I will try.
"izin vermek" literally "permission to give" can be translated as: let, allow, permit etc.
It is by default used with dative, so you might as well memorize it as: -e izin vermek.
Instead of the dative, "için" is also usable. As in "To give permission for something to happen", it depends on the style and the flow of the sentence. The meaning is the same, though.
So in this sentence, the main verb is: "izin vermiyorum" = I'm not allowing
Do we have a word that looks like it's in the dative case? Yep, "izin vermesine".
izin vermek = to give permission
izin verme = giving permission
izin vermesi = hisher/its giving permission
izin vermesine = to his/her/its giving permission
But whose? This is in the 3rd person possessive form, so do we have any genitive present in the sentence? Yep, the very first word is in genitive: Annenin. That's Annen (your mother) + in ('s)
So the sentence is:
Annenin (…) izin vermesine izin vermiyorum.
I'm not allowing your mother's giving-permission (…). = I am not allowing your mother to give permission (…)
Sana bira içmek için = to you for drinking beer
→ I am not allowing your mother to give permission to you for drinking beer.
Or in a more aesthetic way:
→ I am not letting your mother give you permission to drink beer.
Does it make sense?