Is this translation correct using subjunctive?

[saraajackson1]

Because of censorship in Cuba, my father was not allowed to look at American newspapers.

Debido a la censura en Cuba, mi padre no se le permitía mirar los periódicos estadounidenses.

[Rusty]

Welcome to the forums!

Other than one mistake, your sentence is fine, but the subjunctive mood doesn't appear anywhere in it (nor should it).
(The mistake is in the usage of the indirect object.)

Did you have a question about the subjunctive mood or the imperfect indicative tense (the only verb in your sentence is cast in that tense)?

[saraajackson1]

Where is the indirect object mistake? And how can I fix it?

[aleCcowaN]

mi padre no se permitía (he didn't allow himself)
... mi padre no se le permitía (he was not allowed)

but something is still missing

[Rusty]

It may help you to look at the verb using a pattern that shows how it's used, like this:
permitirle algo a alguien

The verb takes an indirect object, represented by the indirect object pronoun le. What is permitted is represented by the direct object algo. The indirect object 'a alguien' reminds us that the indirect object needs to be clarified when the indirect object pronoun is in the third person. Look carefully at the model and you'll discover what you've missed. ;)