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How to convince your parents you're sick
Show and explain your symptoms.
If you're actually sick but your parents are skeptical, the best way to convince them is to concretely show how sick you are. By being forthcoming with your symptoms, you show that you have nothing to hide, making them more likely to trust you.
You don't want to emphasize on them too much, just make your parents aware. For example, if you've had to wake up in the middle of the night to vomit, it's important to mention that.