I started having trouble making myself go out of the house a few years ago, so I asked my Dr for a referral to a therapist and my father agreed to drive me to the sessions every week - your parents might be equally supportive, you never know. My father never wanted to know me, only my brother, but he still stepped up to the plate and helped me out when I needed it. Tell your folks, they might be better about it than you expect.
I know this doesn't suggest how to tell your parents, only that it might go well. I just approached my father and let him know I thought I had developed an anxiety disorder that was making it hard for me to leave the house, then I asked for what I wanted, but I know some people have a harder time with this than I do. My father had noticed I was having worse attendance at work and was wondering what the problem was, so he was kind of relieved to know what was going on with me. What can your folks do worse than just say "no, we aren't able to do that"? Then you will be no worse off than you are right now and will not have to worry about telling them any more, so you'll actually be better off.
We here in Eastern Massachusetts have a handicapped transportation service that my Dr filled out a form for me to qualify for, but you'd still have to propel yourself out of the door to the car or van they send you. Still, the driver comes up to the door and knocks, so you'd just have to answer the door and you'd have a "safe person" to accompany you to and from the car or van. I'd ask mom & dad first, though, cuz you have to wait awhile for your pickup with the HP transport and mom/dad will get you back home safe and sound much sooner after your appt.