What are the liabilities if you decide to become a volunteer caregiver of your parent with mild dementia?

I'd discuss any legal issues with an attorney in your jurisdiction. I might inquire what happens if what you think is mild dementia, so you leave them alone, then they wander away, get hurt, etc. I'd inquire if once you take on their care, you may be limited in just walking away, without getting them proper replacement care.

I don't know of any liabilities, but there is the risk that you will get in over your head after a while when care turns into 24/7 and then you risk becoming a statistic for 40% of caregivers die before the person they are caring for.