well when u get fb u can interact with ur friends,and u only have to give the personal information u want to give. Furthermore u can make that only ur friends have acces to ur data or pics so its actually pretty safe. Strangers can only see ur pic and dont have to make a pic of urself on ur profile.I never had any problem with fb and I ve never been molested by other people.
My parents were just like yours. Just ask them to check it out for you! If they don't, ask some of your friends with a facebook to write down reasons why. :)
There are privacy setting on facebook so only the people you friend are able to see your profile. It's completely safe if you don't add people you don't know!