I think it would be smart to know about all of the tax and financial planning issues first. This would include your parents' assets, liabilities, income, expenses, goals, and risk profile. Then make a plan. Then execute on the plan.
You can get your parents' information from them; hopefully they know it all already. You can investigate the financial impacts of assisted living - check out prices of places, see what their home might sell for if they own it, find out if they have a mortgage on it, etc.
Learning about the tax and financial planning stuff can be complex. Personally I DIY 99% of it by reading forums and IRS tax publications and books and blog posts and other financial resources and thinking about it a lot and asking a lot of questions over many years. That's one route.
You could also hire it out to an advisor. You've identified some problems with that approach. Cost, control, tax, and transparency issues exist. But you don't have to learn and you have someone to blame if you want.
The third way is to find an advisor who will help you on a one-time or periodic basis. Their per hour charges may seem high, but if you do most of the work yourself and are organized with your questions, you can get a lot figured out in a short period of time. My 1% non-DIY is a partner in a local, well-respected CPA firm who understands about tax planning pretty well and seems smart to me. He charges $250 an hour I think, so we buy an hour of his time periodically. In addition to a tax-planning-style CPA, you could also perhaps look for an estate planning attorney with a tax focus, or maybe a fee-only financial planner; although the latter often seems to be an insurance or investment salesperson in disguise and will probably just give you a fancy color printout from some generic software.
As for one of your last questions, it's difficult to figure out all the stuff you don't know yet but should. The only way I've handled that is to read, learn, investigate, ask questions, read some more, until the point where I can skim an article and say, "Oh right, I know about step up in basis", or "Oooh, they got that point wrong about capital loss carryforwards", or "I know the answer to this poster's question so I'll give it", or "Oh right, I don't really know that much about charitable remainder unitrusts, but I know enough to know I don't need one", or "Oh, I don't know about this but I know where to go to read more about it and figure it out". (By the way, the same is true in any knowledge domain - auto repair, marine biology, human relationships, history, physics, etc.)
