From the post, it sounds as if you, your husband and his parents have all developed expectations that their "having a grandparental relationship with the kids" means "babysitting the kids overnight or spending substantial time alone with them." The in-laws, and you as well, need to change this expectation, and it won't be easy at first, because your MIL seems to feel it's fine for her to be in charge of the kids (meaning she takes them wherever SHE feels comfortable) and you are not fine with it. Don't worry about whether others say you're overprotective or you're not protective enough. Go with your guts. But she is going to be hurt and upset and maybe angry, so your husband -- not you, your husband, their adult son -- needs to navigate this with compassion but also firmness.
That means you stop talking and asking. You say, "What other than talking about it can we do?" You stop expecting MIL/FIL to take the kids, and you go with them on any visit, and you and husband drive. Inconvenient, yes, but necessary. You have tried the talking and explaining and you know it does not work in the way you want ; stop repeating a "solution" that doesn't work. You also know that FIL has a serious medical condition and should NOT be driving your kids (and probably not driving at all).
Start saying "Sorry, no, we're busy then" to all overnights. Start going with the kids to see them (which your husband should do anyway, because frankly, if his dad has Parkinson's, MIL needs a break from FIL and your husband can spend time helping out around their house). Your kids can still see them plenty:
You all go over to visit and you and husband stay (doesn't have to be overnight). Or husband takes just one kid over to visit and stays (good time for MIL/FIL to get to know the kids as individuals). Or you take just one kid over, etc.
Invite the grandparents to more of the kids' school plays, recitals, dance events if that's their thing, sports games etc. -- not every single time, but frequently enough that MIL/FIL feel involved. Have them over to dinner or take them out. Drive them and the kids to do an activity -- mini golf, a movie, the park, whatever -- and drop them off and come back a bit later. In other words: You and your husband will have to put aside the idea that MIL/FIL can be alone with the kids for more than the span of an activity, and give up the idea of any babysitting. You know it's the only way to go.
To be really blunt: My own FIL had Parkinson's. It is very tough as it progresses, and it will progress. Your MIL will increasingly need support and help if FIL remains in their home, and she needs to focus on him without any thought that at any time she is responsible for the kids as well. And your husband and you are right that FIL should never, ever drive the kids (or be alone with them -- does your 11-year-old have the presence of mind and experience to know what to do if Grandpa passes out or falls and Grandma's not there?).
Be ready for the blowback or at least the teary "You don't trust me" comments but do not address them. Your husband has to just smile and persevere with the new way you do things: "Sally has a game this Saturday -- Wife will pick you up at 11 and we'll all go to lunch and then the game! How about it?" Make iappen, and stop talking and asking MIL to change. She won't. She does not see her old neighborhood as "bad" and your talking won't change that. I would focus much more on spending time with FIL before he's no longer able to do things with the kids (been there myself).