Amazon Prime is $99 for the year. It includes free two-day shipping on Prime items and you don't have to have the $35 minimum order to do it. You could order a $4 Prime item and it would be shipped alone. Not all the shows to stream on Amazon are on Prime, but you could go to Amazon and search for things you like on streaming and see which ones are. Not sure about parental controls. How we handled it was that I added things to our watchlist and the kids could pick from what I had added. It was good for us the past two years, but we've decided we aren't ordering enough from Amazon to warrant the $99 cost, and there isn't enough left on streaming for free that we are interested in (we're pretty picky, lol). They do change up the streaming selections so things come and go.
Do you already have Netflix or have you tried that? You can try it for a month free if not. We're doing that for now (and had done it prior to the Amazon thing).