Do I have the right to refuse to do unhealthy things they request?

I suppose if you want to keep getting paid you’ll humor him. Who pays you? Where are you living? Is stepmom disabled as well?

Too bad your his daughter. I have said this a number of times, you cannot make him do what he doesn't want to. Your in his home, his rules. It doesn't help to harp at them. Just do what you are hired to do for him. You are not there for her, right. She can get her own mountain dew. Maybe you can compromise.

Good luck with trying to supply only healthy items. My 96 year old mother survives on eating doughnuts and sugary ice tea and about 4 oz of protein daily from her healthy, dietician prepared meals on wheels plan. The nice little spread of whole milk, fruit cup or salad, veggie, carbohydrate and most of the rather large portion of meat gets throw down the garbage disposal after it is left on the counter overnight. Most would say "let them eat what they want to eat". But, as she becomes constipated from her food choices the natural recommendation would be the exact foods that she consistently refuses to eat. I switched her tea to sugarless and I add Benefiber when I open the containers for her. I give her individually wrapped prunes every morning with her daily pills. Once a week, when she is at my home, I prepare her a tiny meal and desert and she will usually eat it all. Even the microscopic vegetal offering. If you are sincere in trying to help them, you need to come up with ways to make tiny little adjustments (most of which will be vehemently rejected). If that kind of flexibility is not in your nature, rethink your choice to be his caregiver.

When you have close, even familial ties to someone you’re caregiving for, particularly a parent whom you’ve obeyed all your life, it’s understandable that you’d have a very difficult time suddenly becoming the one who says no to them, even if it’s for their own good. They don’t realize that. You are his daughter and you listen to him—to his way of thinking.  What will your father do if you refuse? Yell? Leave the room. Same with stepmom and her soda addiction. They cannot force you out to the store. You need to have the power to say no, which they haven’t given you. They haven’t hired a caregiver. They’ve hired a lackey. 

It took me a very long time to develop a thick skin with my husband. When I feel like I’m about to explode I tell him “Ignoring you now.” And leave the room. 

Two choices. Re-draw the caregiver agreement. Put rules into effect. You aren’t giving care if you’re out at the store all the time buying stuff. And, are you also taking care of stepmom? Is that also in the agreement? 

Second choice. Be vigilant and determined to save as much money as you can and move out on your own. If you want to be a caregiver, apply at a facility. They often do their own training at no cost to the student and then hire you to work for them. Offer to research caregivers for your dad and his wife. I can almost guarantee they will “behave” better for a stranger.  My husband sure did. He called her “Sarge”, affectionately and still wants to visit her at rehab. 

There is absolutely no shame in admitting that caregiving for your dad and stepmom isn’t going to work, especially in the capacity of an employee. When money changes hands, everything changes. If you decide to strike out on your own, be honest with them. Tell them you thought you could do it but you can’t. Be kind but firm. And good luck. Let us know how it works out.