Not enough information provided to help. Did they search through your belongings? That's a bit of a violation of your privacy, unless you've given them a reason (drugs, theft, etc). Were you sloppy in hiding things? That brings consequences, too.
First, as a parent, I never violated any of my kids personal spaces, including all of their rooms. That's the privacy thing. We're all entitled to it. Now that they seem to have violated that, and found something "sexual", they should back off, knowing that interfering with your sexuality is off limits. Each to their own. Since their first impression might be that you have a medical problem, maybe you should let them know right off, that is not the issue. This is a choice issue. You feel safe, comforted, etc.
If you dare, you bring them to somewhere like adisc, to learn more, if they're persistent in knowing what it's all about. There are plenty of places on the internet, related to ABDL that you WOULD NOT want them to go to, so this is certainly tame enough that they can learn what they need to know, and leave. Again, this is your thing, not theirs, so a well adjusted parent will realize that you are not harming yourself, your not searching for a sex partner, etc., and they should honor your privacy. I considered telling my parents, at certain points in my life, but decided it just wasn't anyone's business by mine, and my partner's. Enough said.
There's no putting the toothpaste back in the tube, so you need to have a frank discussion. We see these messages all the time, and it makes me wonder how you let it happen, and perhaps why. Some would say you were asking for it, by not hiding your stuff well enough, and others would say, it's NOTB, but parents are parents, and believe me, they're probably just worried about where this stuff is headed. Personally, had I ever came across anything sexual in my kid's stuff, I would have just had a chuckle, and left it alone. Let them develop on their own. In this case, they might be considering blaming themselves, which I think is a waste. My issues developed without any intervention on my parent's part, to the best of my knowledge. It was just my own damn diaper brain, wanting to be like the other little kids, still in diapers, when my mom had just had enough of them, and was sure I was ready (I can assure you, I was not, even though I could keep things dry).
Best of luck in sorting it out. My suggestions are to find better hiding places, or, at this point, just let them know, you will make every attempt at keeping it private. If you seem to think that means you can run around the house in just a diaper, let me tell you, that's taking it way too far! Once you find your own apartment, or home, you can very well do what you want in your own castle, but, right now, you're living in their castle, and you need to abide by their rules, EVEN if it means no diapers, until you move out. It would mean they were sticking their noses in your business, but, you made the issue pop up, by not hiding things from them, a bit better, or by not having a discussion, prior to their figuring it out for themselves. Either way, you all have to work on a solution, and for them, the best solution might be to forget they ever saw anything, and to never hear you utter another word about it. Eventually, you'll have your own place. At your age, I was in the AF, and had to put off wearing for several years, until I was in a more suitable place in life to live out my diaper dreams. There is something about kids these days, and the entitlement society, as if you're entitled to whatever you want, but that's a fallacy. You're entitled to what you earn. You want respect, earn it. You want money, earn it. Etc.
Hope you can take one or two tidbits from all of this. Search out other threads on people whose parents have made the discovery, and learn from their mistakes. You're not alone. As a matter of fact, you're among a crowd of others that have left their guard down, and had to pay some consequence, sometimes trivial, and sometimes life changing. Consider all of this in all of your other relationships (with partners, bosses, the local police, online predators, etc.).
Good luck!
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Some very good replies above mine, too! See, the community cares!