Technically speaking, the answer is no - they cannot search your belongings that are clearly yours.
They would be permitted to enter the room and inspect what is there since they own the property, but not your stuff since you are a legal adult.
Even with this being the case, you don't have many options here. Actually, you have three.
1) You could try to invoke legal remedies (sue them), but it would never go anywhere in a court.
2) You can stay and deal with it, or
3) You can move out.
Option #1 won't help you at all, so just eliminate that. Just think to yourself, "Should I stay or should I go?" And decide from there.
EDIT: Thumbs down all you want. The owners of a property do not have the right to search through belongings of adults without permission. This is why you can refuse, for instance, to allow a store to look through your bag if the alarm goes off. The asker of the question is over 18, making the person an adult. The parents can enter and inspect, but cannot legally search through private property regardless of relationship. That is actual law - disagree all you want.