Accept That No One Likes Criticism
No one likes to be told that they are wrong. Yet, every single person has, at one time or another, been criticized for something. Realize that it's a normal part of life. You might receive some harsh criticism, but it doesn't mean that you are dumb, or incompetent.
Stand Up For Yourself
If an elder or family member becomes abusive, you can calmly say you won't be treated like that and walk away. Once they see they aren't getting the desired outcome -- to get you riled up -- they might "cease and desist." Another way to stand up for yourself is to remove yourself from the situation. Find home care for your parent. Having a stranger who isn't quite as caring, or familiar, might make the elder realize how much they want and need your help. Let the family member step in to the role of caregiving. Once they've experienced how hard it is first-hand, and walked in your shoes, they likely won't be nearly as critical.
Smile
Smiling, even a false smile, can help us relax. It creates a more positive vibration and diffuses the situation. It will definitely help you psychologically. It's hard to be upset or hurl back insults when you're smiling.
If you can learn how to cope with criticism, you can eliminate the stress and anxiety over what people think, and be happier and healthier.