Hi carmy,
Thank you for reaching out! This is entirely possible because SSRIs are known to impact sexual functioning, especially with dryness. While clitoral soreness can come from dryness, it can also be from the form of masturbation that you use. I understand your pain completely because I have been in a similar situation, so the best thing I can do right now is to tell you that you are not alone. With all of this in mind, I think looking into options for lube might be a great start to alleviating this issue. I know you said that you aren't able to get it without your parents knowing, so we have a good article that may help you out:
Lube 101: A Slick Little Primer. Also, depending on if your coconut oil has any additives, you should be fine to use it!