God bless you! I am proud of you for caring so much about your soul and wanting to go to confession! It is true that you did not commit a mortal sin if you did not know your sin was wrong. The conditions for a sin being a mortal sin are:
1-it has to be grave matter (yes, it was)
2-you have to know it is wrong (you did not)
3-you have to commit the sin freely (it appears as if you did)
I would still do whatever I could to get to confession though. You are going to be tempted again (and again, and again…) to this and other sexual sins over the next few years (well, temptations will always be there, but they are often more difficult in the teen years), and you need God’s grace to help you resist these temptations and to make the right decisions instead. Even if you don’t commit any mortal sins again (masturbation would be a mortal sin for you now that you know better), it would still be good to try to go to confession regularly. The Church doesn’t require this - but many in the Church recommend doing to confession as often as once a month.
Are your parents arguing with you about the value of going to confession, or are they forbidding you to go? If they haven’t forbidden you to go, I would encourage you to ask again. Here’s the thing about your parents’ argument - for VENIAL sin, you don’t absolutely have to go to confession. You can ask God to forgive you. But even for venial sin, confession is recommended by the Church. You receive grace when you are absolved from your sins by a priest, and that extra grace helps you to resist temptation in the future. What’s so bad about receiving a little extra grace? That said, arguing theology with them may be difficult, and isn’t necessarily going to get you what you really need, which is the sacrament of confession. So I would just tell them that you understand what they are saying, but that you really want to go to confession anyway, and ask them if they would please take you to the church on Saturday afternoon for about 15 minutes (if you get there early, chances are you won’t take longer than that) . A few ways to make this convenient would be to suggest that you just go to Mass together as a family on Saturday evening that day, after confession. Or if you have any errands to run, to swing by the church on the way to the errands.