If you tutor long enough, especially if you tutor long-term for students, these questions are bound to come up one day: questions on religion, sex, gender issues, abortion, etc., the untouchable questions. Being a full-time tutor myself, they come up with frightening regularity, with religious questions coming up at least once a month and sexuality questions a few times a year. As such, I have learned it is best to get an idea of your approach before the questions come.
First and foremost, I believe it is important to include parents in any dialogue. I do not mean going to get the parents of the child. Rather, I mean to open the conversation with asking about the student's parents. For example, "Have you asked your parents about their opinions?" or more directly "What did your parents tell you about this?" Find out if the student has reached out to their parents or not. If the student has not, find out why; if the student has asked them, ask what the parents said.
At this point, you need to decide how much you want to get into the question. One option is indeed to brush off the question, telling the student to just ask their parents. Although it might be the politically correct option, I generally advise against that approach. If the student is coming to you, there is generally a good reason since these questions rarely come easy to them too. If the student has indeed talked to their parent and the parent answered, my response is generally to affirm what the parent said and get back to tutoring. If the student did not ask their parents, generally I advise the student to ask them first and if it does not work, I can touch on the subject after they try to talk to their parents.
If the parents are out of the equation, such as the student being blown off or another situation that stops them from being able to talk to their parents, then you have to make a difficult decision. Either you can dodge the subject too or you can answer it to a point. Neither is generally appealing, but for the sake of the student, I generally lean toward the latter option. If a student cannot talk to their parents and you shut them down too, they will very likely keep asking others until someone answers; I would rather be the one providing the answers than who else they may possibly find to answer. If you are going to shut down a question, be careful with how you do it. For example, I was recently asked about what are the aspects involved in being a transgender; my response was that he probably did not want to know what all it involves, but if he was still curious later on, I would rather him ask me than try to find the answers elsewhere (besides his parents of course). Coming from that angle, I was able to turn down the question without closing it completely; a closed door automatically invites curiosity about what's behind it, whereas a door that has been cautioned still arises curiosity but not to the same degree, especially when it is known that if the curiosity does not leave, it will be answered later. Other times, you may find yourself having to answer the questions. If that is the case, I advise working from your knowledge of the parents to try to answer similarly to how they would they would respond with your own knowledge. Try to keep the answers simple and generally go only as deep as you have to; you can go into the basics and generally leave at that. The point is to satisfy the curiosity without going too far, especially with sexuality questions. With religious questions, an extra option is to ask questions yourself to make the student think about their question in a new perspective or offer ways that they can research their own answers.
In short, as a tutor, don't simply avoid the tough questions. While they are unpleasant, if a student is turning to you, they might not have anyone else safe left to turn to. When you talk to them, try to go along what their parents said or would say. And, try to keep answers to satisfy curiosity or for religious questions, feel free to ask questions and give ideas for research. The big thing is to not close the door; you can downplay questions to some degree, but a student who desires to know an answer will find it, especially in this Internet age.
As for parents, please be open to your children. Offer to discuss these topics with them. Perhaps set up a time to talk them over. I see far too many times that kids have to turn elsewhere for answers.
Please feel free to share your own thoughts/ideas.