You seem to be having a pretty good idea of what goes where. Werden, being the finite verb, needs to occupy the second position, and sends erlauben to the end of the sentence. The infinitive construction, however, is required to be even behind the verb at the end, in the so-called Nachfeld.
There are two possible ways of saying to watch TV; you seem to be mixing them together creating an uncommon way of phrasing it. You can either say Fernsehen schauen or fernsehen. Together with the infinitive-zu, these turn into Fernsehen zu schauen (no questions there, usually) or fern zu sehen (that one’s a separable verb).
Instead of fernsehen you can also say fernschauen (fern zu schauen). Which variant is preferred depends on the region.
And finally, although that one’s really minor and not entirely relevant to the question, jemandem etwas erlauben requires a dative object for the person receiving (or not) permission.
So one way of wording the sentence would be:
Meine Eltern werden mir nicht erlauben, fern zu sehen.
But German has — as you probably know — a very flexible word order. The following sentences are also correct and only represent a subset of all possible ones:
Fern zu sehen werden mir meine Eltern nicht erlauben.
Mir werden meine Eltern nicht erlauben, fern zu sehen.
And of course, the same excersize can be excersized with Fernsehen zu schauen:
Meine Eltern werden mir nicht erlauben, Fernsehen zu schauen.
Fernsehen zu schauen werden mir meine Eltern nicht erlauben.
Mir werden meine Eltern nicht erlauben, Fernsehen zu schauen.
Unfortunately, having the infinitive right before the erlauben of the future tense is not possible according to my sprachgefuhl.
Meine Eltern werden mir nicht fern zu sehen erlauben.