It sounds like you need to set some boundaries with them. Tell them what you can be available for, explain that if you have plans you can’t just drop them etc.
And as Simon has said you could maybe just not be available for a bit.
It’s a tough dilemma because I assume your parents are elderly and mum doesn’t drive. And then you have wherever your dad’s currently mental capacity is at just now. So you know you can’t completely abandon them, and they are your parents after all. But they are not your children either! And your dad does need to relearn to do things for himself for the sake of his brain’s retraining; for the sake of his own health and wellbeing.
Couldn’t you meet him halfway by insisting all nonessential driving he has to make his own way, and just stick your guns. But do it gradually to wean him off being dependant on you.
Just as simple examples, unless you have anything more specific you’d like to add.
Lets just say he wants you to take him down to the library. Just say NO! Not today, you’ll have to find your own way today. You don’t even need to give them a reason why you can’t, just be firm that you are not available today.
or I can’t today because I won’t be here…no need to say where you’ll be or make something up.
or I can’t today but I’ll be free on Tuesday, sort of thing.
or have few set times/days in the week to do some running around for them.
Not sure if you are a parent or not, but I’m guess not or your really spoiling your kids 
I’m just joking now. But that’s how you’re going to have to approach this. Are they becoming too spoilt because you are not being strong enough to say no to your parents. And you’re doing them no favours. And it is hard to say no to your parents, especially if they have been the kind of parents who have been there you all your life. But you are also an adult, who needs some life of his own. There’s a difference between being a caring, helpful son and being a slave to your parents…that one’s not good for your own mental health.
So you need to be firm but fair and stick to your guns.
Boundries . . . that’s the word I was looking for 
Rules and Boundaries, that’s just what you need to set out @RobCharles1981 
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if he doesn’t feel capable I think you must go with that , usually people say they are ok to drive when in fact they absolutely shouldn’t.
I understand that you feel you will become a slave,I don’t have an answer to that
problem. most of us feel guilty for being a burden.