I cooked chicken in a pot, then I took the chicken out and made a parve sauce in that pot. I didn’t clean the pot after taking out the chicken, and I think there were still a few small pieces left. I then tasted the sauce. Do I have to wait 6 hours to eat milk?

You do not have to wait 6 hours. Since you didn’t intend to leave the chicken in the pot when making the sauce, rather just out of convenience you didn’t bother cleaning the pot, the sauce does not get the status of being meat to make you wait 6 hours. This is especially true if there were only a few pieces of chicken, as you described, which are בטל בששים.

ש”ך יו”ד סי’ פט ס”ק יט