This happens to many people, and they get through it. Depression is horrible, but please realize that there will be better times ahead. It gets better, I promise you.
You absolutely do need to take a leave of absence, for health reasons. I am sorry that your parents are not being supportive. It’s good that you understand that their failure to acknowledge your depression can be a cultural issue. It’s very common; I’ve even seen it in physician parents who absolutely would recognize and recommend treatment for their patients with depression, but just cannot see that their child is having it, and needs help.
I’m sorry that counseling services through your school are so bad. Your therapist is flat out wrong. You need to do an end-run around this. You can get access to medication and other help by going for a medical evaluation at student health - say you’re extremely tired, sleeping 16 hours a day, and think you might have mono, in order to get the appointment, then when you’re in, reveal the depression, and how bad things are. They’ll get you in with psychiatry asap. Hopefully, they’ll also take a good history and do some directed bloodwork, because sometimes physical issues can be the cause of the depression - things like hypothyroidism, severe anemia, and more - these are the purview of the internist or adolescent med doctor. Some of the antidepressants do kick in more quickly than others - hopefully you can discuss this with the psychiatrist. It’s also very common that the primary doc will start you on an antidepressant right away.
I would also get an appointment with disability services, because you are possibly going to need accommodations to finish up the semester, if at all possible - things like being able to turn in work late, retake exams, switch into online sections and then take an incomplete, and finish them up online later on.
Is there any trusted friend who can help you with getting these appointments? It’s so hard to fight the battle to get help, when you’re having major depression.
You need help, and right away. Please call for an emergency appointment at student health now, or first thing tomorrow morning. Also, is there anyone else you can call for support right now? A sibling? A trusted friend? You need to reach out for support. You cannot handle this on your own.