I can understand your fear of meds. I took a Trazodone once to see if it would help me sleep. I laid awake obsessing about a drug in my brain. I have finally gotten to the point that I can take an anti-depressant without obsessing about what it is doing to my brain. However, I suspect that your situation is scarier because it involves more drugs than just one.
I researched my drug pretty well so that I would know the side effects. Have you researched your drug? (If you can give yourself side effects by researching the drug, proceed with caution.) I tend to feel less nervous when I know things. If that is the case for you, then it would be a good idea to go to the library and read the Physician's Desk Reference, Pill book or something similar about your medication.
You might also consider starting with the most important drug first. Can you take a dose when you know that your parent's will be home and during normal office hours. Once you have done that a few times, then you could try to adjust it towards what your doctor has recommended. (Prozac will start out with some annoying side effects. But, if you keep it up, you will probably see then decrease or go away in a few weeks.) I am on a different anti-depressant of the same classification. I would talk to your doctor if you decide to try to gradually work your way towards your doctor's orders to make sure that it is okey to try it.
As for the hospital, I have never stayed in a psychiatric hospital and hope to never do so. I think that it might be hard to convince an insurance company of the necessity of such a stay. It is hard to convince an insurance company to okey a hospital stay unless you are suicidal, unable to care for self, a danger to someone else or some other very serious situation. I hope you find a way to get yourself into feeling better without a hospital stay. However, if it is needed, go for it. I would definately want to pre-plan it as the above posting recommends.