Can parents throw out belongings you have kept in their house?
Legally, it depends :-). In particular, it depends on the agreement you had with your parents, i.e. whether they promised they would keep your belongings. However, even if they promised nothing, they would probably have been required to warn you they were going to throw them away, and give you time to get them if you wanted to.
Unfortunately, that has not happened. At this point, legally, the only thing you could do would be to sue for damages. However, to successfully sue them, you would have to convince a court that you left certain possessions with them, and that they threw them out without telling you - both of which might be hard to prove. Even then, you would probably only get back the present value of your belongings, which for used clothing and other personal items will probably not be much. So in practice it might not make sense to sue (totally ignoring that it would probably also sour your relation with your parents).
I asked my dad and he said I wasn't 19 when I got most of those
things so they aren't mine. Is that true? Does it matter if they were
gifted to me?
No, and yes. In general, minors can and do own property, so your being a minor when you got these things does not automatically mean they do not belong to you. However, if your parents bought them for you, it might not be clear whether they were intende as a gift to you, or just loaned (e.g. with the intention to later pass them on to your younger sibling or friends). However, if they clearly were gifted (e.g. birthday present from your aunt), they belong to you.