I'm not the angry one, it's my mom. I living within walking distance of my parents. I take both parents to their doctor appointment. My Mom becomes angry at me even when I haven't been there. She has my Dad, who has lung cancer, cancels her appointments. I don't know how to handle her angry outbursts.
If no one is PoA for your parents, this is another issue. If your dad has lung cancer but is competent, ask him if he'd like to make you his PoA so that you can continue to advocate for him. Take him out of the home temporarily so that your mother doesn't know what you're doing. Then, if your father makes you his PoA, maybe your mom will consider doing it also.
If you aren't his PoA, then maybe take him out of the home for a day or 2 to get him to appointments. Have him request the Medical Representative form (HIPAA) to make you his MR. Make sure the clinic knows that your mom has the beginnings of dementia and to not cancel your father's appointments unless it comes from you. You will need to call the clinic in the days leading up to his appointment to make sure it isn't getting cancelled. Maybe erase the clinic namen and number from her devices or phone book to make it harder for her to undermine your efforts.
Also, even without POA, you can share information about her behavior her doctor.
Tough to arrange this if she's uncooperative and can tell someone who the president is and the day of the week.
If she acts out, becomes aggressive, you can call 911 and say that you feel she is a danger to herself and others. You get her remanded for a psychiatric hold and go from there.
A consultation with a good eldercare attorney could also be a good move. S/he might have some other ideas based on your state