What do you do if a parent goes to the bathroom on herself if she doesn't get her way while you're out in public?

If you are willing, carry a box of disposable plastic latex-free gloves in your car, whether it be where you can reach it in the back seat or in the trunk. The trunk might be the better spot, and it might also be a good idea to carry extra DEPENDS in the trunk and keep a couple in her purse (in an empty grocery bag), so that if you are out, you can help change her.
If she continues to poop and pee, see a geriatric psychologist (psychiatrist) or have some tests done in the drs. office (primary care physician). It could be from a UTI--Urinary Tract Infection (they can play havoc with a person's body and also, how they act).

she wears depends but she carries the stank smell with her while we are out and would be in another state visitin family or business

I'd say to your parent if you do that one more time, you're going to have to start wearing disposable underwear (e.g. Depends) or you can "nip it in the bud" right now and just get some disposable underwear (e.g. Depends) for your next outing and give it to her at that time, & tell her, "Before we go out, please put these on & explain why." Or not explain, but I found my MIL always felt more inclined to go along with what I was suggesting if I explained to her the reason behind it. Even though my SsIL (Sisters-in-law) said to me "TMI". TMI stands for "Too Much Information". They always felt it wasn't necessary to explain because she wouldn't understand it or wouldn't remember it. She has Alzheimer's Disease (AD). ...but everytime there was an explanation that I gave her, she thanked me for it afterward.

Well, they decided to put her in a nursing home and told my husband of their decision. ...and she was living here. They'll have to just deal with it.