My career background is directly related to this issue, and I can tell you from professional knowledge that:
1. HOAs definitely have the right to enforce their bylaws, and your parents signed those documents and were provided the documents when they closed on the home. If they want to change the rules, they should go to the board and discuss the issue. Ignorance isn't an excuse, even if the rules are arbitrary and ridiculous.
2. Many/most cities would define detailing for money in your driveway as a "Home Occupation", which requires licensing and adherence to rules (e.g. no/limited signage only using a certain portion of the house for the business, not having customer traffic, etc.). The reason for this is to keep your neighbor from putting out signage and degrading your residential community with activity that the zoning doesn't support. This protects property values.
I think people often cry "property rights!!!" without thinking about their NEIGHBORS' rights. I love detailing my cars, but I'm very aware that the buffer is noisy, and if I were regularly detailing cars in my driveway, I could see how it would piss my neighbors off after hearing 8 hours of buffer whine for days on end.
My advice is that you should take the high ground on this one. Go over and introduce yourself to your neighbor, and apologize for any inconvenience you've caused. Explain that you detail a car or two every week in order to help you get through school/earn spending money/whatever, and ask her what you can do to make it more tolerable to her and her family. Even if she's a complete PIA and refuses to even have the discussion, at least you'll know you've tried.
After all of that, go to a HOA meeting, explain yourself, ask for their consideration of setting specific limits on your operation (e.g. certain days/hours, only using a buffer in the garage with the door closed, etc.), and apologize for causing the issue. Everyone demonizes HOAs, but you can be sure that they're as pissed about having to be your nanny as you are for having them do it. Just be professional and respectful, and don't go with a chip on your shoulder. They'll be impressed by your character, and just might work something out with you.
Good luck, and I hope you can reach resolution with the neighbors and HOA and continue the good work.
(By the way, I'll help for free if you need assistance with the Home Occupation license - just send me a PM and tell me what City you're in)
**UPDATE: I see you're in Irvine. Home Occupations are licensed. Here's the application w/criteria for same. If you're able to get through step 1 with the HOA, then I would go ahead and get this license so you don't have to worry about being fined by the City:
http://www.cityofirvine.org/civica/f...sp?BlobID=7053