I don't think there's one thing we can tell you regarding how to resolve this problem, because every parenting situation is different. The best way to deal with the situation with your parents is not the best way to deal with the situation if I were your parent. We're all different. That being said, there's a few general approaches which have a proven track record.
I think the first step is to think critically about the "big time" punishment, and how big it really is. You haven't stated what the punishment is, so we have to take your word on it, but really take a look at how big it really is. Teenagers often blow things like this way out of proportion (source: I was a teenager once). You're going to be 18 soon, and you'll find that the punishments that you may face are much more dire than anything a parent can do, and you may face those punishments even if innocent. We don't like to admit it, but innocent people do get punished.
You have to think critically about this punishment because the severity of the punishment affects how hard you should work to be free of it. If it's actually a small thing that you've blown up out of proportion, it's better to show to your parents that you can shrug off the punishment, without admitting guilt, and get on with life. That's a good skill to have. On the other hand, if it is indeed as big as you think it is, to the point where it is going to have a material effect on your life down the road, then by all means, spend the effort needed to respond to the punishment and nullify it.
Try taking the emotion out of your attitude about the punishment, analyze it, then put the emotion back in. If you can analyze it the same both with and without emotion, there's a good chance that it's a good and reliable analysis. If you notice that your opinion shifts when you let your emotions back in, listen to them. Then, take the emotion out again, and try to address what they told you. The back and forth process is slow, but eventually leads you towards a position that you can be comfortable with.
Once you understand how big or small the punishment really is, and how much energy you want to put towards resolving it, now you can look at the players on the field. There's actually 3 entities you can interact with. There's your parents (2 of the entities), and then there's the punishment. The punishment gets isolated because once your parents enacted it, they are obliged to enforce it (to show that punishments mean something). They will certainly have insulated themselves from the punishment. This is important because if you try to tackle the punishment head on, you'll find its a difficult task. The punishment was designed to deal with you, after all! If you're trying to directly convince your parents that you shouldn't be punished because you didn't lie, your parents are going to mentally direct that argument towards the punishment because it's designed to be resistant to appeals like that.
The alternative is to interact with your parents by going around the punishment rather than through it. This is a difficult skill, so don't be surprised if it's hard. Your goal is not to prove your innocence, but to remove the punishment. You can always prove your innocence later (as long as you haven't admitted guilt). You should look at your parents to try to figure out what sorts of things they need to feel in order to want to remove the punishment from you. This isn't a straight forward path of going to them and asking "what do I need to do to remove this punishment?" but more of a soul seeking effort to try to find something that you can do which makes it easier and easier for them to want to remove the punishment.
Look for win/wins. If you look into your parent's hearts, tell them what they want to hear, giving them the illusion of progress, it's called manipulation. But if you look into their hearts, find what they want, and then make it real, then that's called finding the higher ground. As a hint, they want to be able to trust you, and that trust has just been shattered. Acts which help them pick up the pieces and reform that trust will go a long way.