Schools need to be responsible for most of it, and especially how to find reliable information if you're not sure about something.
In an ideal world all parents would be brilliant at it and there would be no need for schools to be involved. But that's not the case.
Sometimes the parents may be very conservative (perhaps religious reasons) and teach their child utter BS to try and put them off. Some may teach them lies, such as about how effective or ineffective certain contraception is. Some may teach them homophobic stuff. Some may simply teach abstinence-only sex education, which has been proven time and time again in the USA to be useless. etc.
Some may be perfectly decent people who want to teach the right things in the right way, but may not very good at teaching that sort of thing. Or they do teach it and miss stuff out.
The schools absolutely must play a role, so that they get taught the correct stuff and know where to go for reliable information if they want to know more (some might come across unreliable and propaganda websites while searching on the internet).
Most parents are probably pretty good at it, but we need schools to make sure everyone has some understanding.