I don't want to invite my dad to my wedding
Is there something wrong with your dad?
Yes, I'm also afraid that he or my mom would embarrass me in public
Your dad being excluded from his daughter's wedding would be really hurtful for him. It also may negatively affect what people think of you too as someone who didn't allow their own father to attend their wedding.
Whatever. I just want to have a respectful existence in my community and co-workers.
It is rude to exclude people unless they are really out of control. You should talk with your parents about the wedding and how you feel, they may be willing to change.