You need to gain your parents' trusts, in the sense that they believe that you are capable of taking care of yourself. My parents let me go on my solo trip because they knew that I was self aware, and could handle tricky situations myself. Another important factor is the place. You might want to start out with a place which is either close to your city of residence, or someplace where you have already travelled to with your family, so that they also have the reassurance of knowing the place.
Have been through this and now my parents are fine with me traveling solo (though are always still worried, just parents thing). I started traveling solo with a group (trek groups) and shared numbers of the organizers, made them talk to the organizers. Eventually, they also gained confidence in me about traveling solo!
Don't force them or argue with them, take it slow! Else take one of you sibling along, even better, one of your parent and get them addicted to travel :P
This is a common issue most of us face. Head to a place familiar to them. Or you could go to a place where you have relatives where you could stay over. If nothing works, tell them you're going for an official trip alone.
Even though many might not approve of my idea.. what I did was I went solo with a trekking group since I was wandering out alone for first time.. after the successful trip I came and told all the details about the trip to my parents.. Seeing how much I enjoyed and was safe during the entire trip now my parents know that I'm responsible enough to take care of myself and won't object while going solo.