Although the main Duolingo website does not have parental controls, we do offer it for teachers and parents via Duolingo for Schools (also free and you don't even need a new account — just use your Duolingo account). From there, a teacher or parent can disable certain vocabulary words for their student, as well as social interactions on the website. If you are interested in doing this, please go to How do I disable social features and mature vocabulary?
It should be noted that Duolingo aims to be a positive and safe environment for learning, and with the help of awesome moderators and a community that is helpful and share a love of learning. Be aware that your student may be missing out if prevented from interacting and learning from native speakers who explain tricky concepts in forums and sentence discussions.
If someone in particular is bothering you, please read "How do I block someone?"
If you see forum abuse and extreme misbehavior, please read "How do I report abuse?"