I suggested to my father, that he and my mother give me Power of Attorney, but he does not want me to have it. What is the con of this?

Does your father understand that a financial POA doesn't take effect immediately? It's only a "just in case" something happens to him. He retains full control until declared incompetent. In our state, we had to have 2 doctors sign off that mom was incompetent before we could take control.

If this does happen, would he rather have you in charge of his money or some stranger appointed by the courts?

CBA: whenever you address "end of life issues" it becomes very unsettling to seniors...although, I wish they'd understand that it is not much fun for the caregiver either! My friend's father wants her to handle their estate after their passing, but refuses to write a will. He wants her to figure out everything after they are gone. She has 6 brothers and sisters. Can you imagine what life will be like for her trying to divide up everything equitably?
Do your parents fully understand what a POA does? Do they know that they need both a medical and a financial POA? If they become incapacitated, and have not given a family member their POA, then other people will be making those decisions for them. If they become mentally incompetent, they cannot even sign a POA and, again, their fates are in "strangers" hands.
Good luck...this is never any fun.

Instead of POA, ask your dad if he'll just put your name on their bank account with them. He keeps control of the checkbook etc. but in case of an emergency you can still act on his behalf.

My Father and Mother made me the executor of their estate and have recently asked me to give them some advice on how to invest their money in order to make more income. So I think they trust me and know that I would not deceive me but I do think there is a "control" factor. My father basically believes that we can deal with the courts and legal system to work it out instead of a POA. He seems to be getting more angry about things that are just advice on things he used to and still does basically handle. I need to really watch what I say in a casual way...thanks for your input.

People from our parent's generation went through the depression and saw fortunes disappear when they trusted banks and other institutions. Really, giving a POA to anyone IS giving up control and trusting, without question, that someone will take care of your life for you.
Or they may be superstitious and think that if they make these arrangements, it means the end of their life.
If a parent has reason to believe that a family member will not represent them well, they have every right to refuse.
A POA is a practical measure, but you really need to find out why they are so hesitant.

did u ask ur father why he doesnt want u to have it ? you should ask him who does he rather be a poa ? there could be a reason why he doesnt want you to handle it .