The only thing I remember not being allowed to watch was Monty Python. It was on PBS at the time. I think my parents were afraid it was all naughty bits
ETA since we are including movies, I was not allowed to go see an R movie until 9th grade - Fast Times At Ridgemont High as it had been "vetted" and there weren't any penises in it. We had a HUGE fight over Porky's, which I still haven't seen uncut. I must mention that most of my classmates had been going to R movies since they were 12 or so. The only other R movie I saw before I was 18 was one of the Friday the 13ths. (I guess they figured I was going to college in a few months so whatever) Given all this, it was truly bizarre to watch "Tales of the City" with my mom when it was on.
I need to add that my parents have never had and still don't have basic cable let alone HBO etc.