Is there an alternative to washing my mom's private areas?

Yes, this really must be done daily. The genital area is one of those parts of the body that need to stay clean, especially the backside. Anywhere that's fleshy and moist is prime territory for bacteria growth, which can cause odors and skin irritation.

Try not get hung up on it. It's best to think of it as if you were washing a baby or child. You just need cleanse the area quickly and be done with it.

There's no need to scrub. Simply wash and rinse the labia and outer vagina and between the buttocks. Use a wipe if you like. It may be easiest to do this while your mom is sitting on the toilet.