Can a power of attorney exist over both parents in a single document?

No, do them as separate documents, since each document is signed and witnessed or notarized. Dad cannot grant you POA over mom; she has to sign it herself. Dad can only grant POA over himself.
Seek Guardianship over mom; she may have been diagnosed and is not telling you what the doctor found. Your siblings and Dad must all agree on who the Guardian will be and agree that Guardianship is needed.

In most (perhaps all) states anyone may seek to have someone placed under a Guardianship. The Court can order an examination if the person does not consent to one.

Because a Guardianship takes away someone's legal rights, there must be a court proceeding in which the person who might be placed under Guardianship is represented by a lawyer, usually someone appointed by the Court. While your siblings and Dad need not all agree on who the Guardian will be and that Guardianship is needed, agreeing on who the Guardian should be instead of fighting over it in court would be less expensive.

Please also note that the person appointed as agent under the Durable [Financial] Power of Attorney and the person appointed as agent under the Medical Power of Attorney should not be the same person. This avoids charges that the cheapest, not the best, course of treatment was chosen. Some spouses name each other for both but given your father's age, this may not be the best course.

I agree with Pamstegma. I would add if you are going to be the caretaker be sure you are the POA and Health advocate and not a sibling. The other good choice is a trusted non-family professional. It reduces problems down the road with relatives.

I agree with those who said No. Dad does not have the right to make the decision for Mom as to who it is. Its rare , but I have seen parents chose different children. If Mom can sign, she has to herself, if she cannot understand, then someone has to go to the court and seeking guardianship, might also be called convership depending on your state

I am going to be 75 and in good health. Only thing stopping my good health is this world. A school bus hit me about three weeks ago. (their fault). Thing like that change my good health. Things out of our control. I have already signed a document on my medical requests and who will care for me and who will make the final decisions. I have even written my obituary so the kids don't have to dig through everything to remember who is this and that. Makes my life a lot more rested. But, I agree with everyone that there should be separate documents. After all, they each will have their own wishes about decisions at the very end.

As many have already said, each parent needs to have capacity to give POA to someone else and they can't be combined. It's unclear whether your mother still has enough decision-capacity left to assign a POA. If she does it with a lawyer they are supposed to check that she has capacity to sign the document. Guardianship is expensive and time consuming, so would be good if you can avoid it.

Regarding your mom refusing to let anyone go to the doctor with her, it is indeed common. I would suggest you contact the doctor though. First, you want to make sure he/she is aware that you think your mother has developed dementia. HIPAA doesn't prevent you from disclosing things to the doctor. Second, you can express your concern that she's refusing to let anyone accompany her, and see if the doctor would be willing to encourage your mom to have someone come to the visits. Sometimes the influence of a doctor can help with this. Lastly, the doctor might be able to encourage her to complete a durable POA for health, assuming the doctor feels she has capacity. Even if she keeps forgetting what she signed, if she's consistent in saying "If I were injured or couldn't make medical decisions, here's who should do it for me," she might be deemed to have enough capacity.

Last but not least, I recommend you review the government FAQs on HIPAA and disclosure to family and friends. Clinicians can exercise judgment and release information if they think a patient has lost capacity to decide on this and if it's in the best interest of the patient.
https://www.hhs.gov/hipaa/for-professionals/faq/disclosures-to-family-and-friends
For instance:
"Q: If the patient is not present or is incapacitated, may a health care provider still share the patient’s health information with family, friends, or others involved in the patient’s care or payment for care?

A: Yes. If the patient is not present or is incapacitated, a health care provider may share the patient’s information with family, friends, or others as long as the health care provider determines, based on professional judgment, that it is in the best interest of the patient."

bluebonnets --
Thanks for mentioning "Please also note that the person appointed as agent under the Durable [Financial] Power of Attorney and the person appointed as agent under the Medical Power of Attorney should not be the same person." I didn't know that. I am 85 and putting all my documents in order.

Just wanted to put in my two cents. In seeking guardianship if the principal is in agreement or is deemed unable to participate in the process - largely due to competency- the need to legal representation and the court process can be waived. The court will however likely want to have independent evaluations done to determine if this is the case. As far as having the same or different person as medical and financial DPOA- this is a matter of opinion. If two separate people can work well together it can be a good call. However, in more adversarial situation having one cook in the kitchen can be a better call.

Each of your parents should prepare and sign what is sometimes called a "durable power of attorney for health care." This document can go by other names, including "Advance Health Care Directive." This document is designed to be effective if the signer has become incompetent and can no longer make his or he own health care decisions.

It is not uncommon for the elder, in this case your mother, to shy away from assistance, e.g. refuses to let anyone, even her spouse, go along to the doctor's appointment(s) with them. And why? Because their minds, sadly, have changed and their decision-making skills are oftentimes non-existent and/or altered if you will and should be left to someone else. I have seen it time and time again.