I am a kidney donor (16 years ago), but I have met, and read about, many liver donors and liver recipients. Living liver donation is a much more serious operation than kidney donation, with a much higher risk of complications and hospital re-admissions. So it is quite wise, I think, to be concerned about the dangers and one's abilities to handle such anxiety. That does NOT make you a bad person! It is admirable that you even THINK about the possibility, because many people (even family members) would not even dream about it. But living liver donation is not for everybody.
Please keep in mind also that you are not the ONLY hope for your parent. I have met many persons who received a liver transplant from a deceased donor and are now doing fine. In some parts of the country the waiting list is long, and not all patients can survive until one becomes available. But you CAN help by encouraging as many people as possible to sign their driver's license or card giving permission for one's organs and tissues to be transplanted after death. Many people do NOT sign, and so many organs are thrown away in the grave or cremation. (Have you yourself and other family members signed your licenses?) The more organs available from deceased donors, the shorter the waiting list.
It may also be important to see if the parent managed to stop drinking. There are people who destroyed their livers through alcohol abuse, received a transplant, and then went right back to drinking and endangering the received liver. There is not much sense (in my opinion) in risking one's life and health to donate an organ if the recipient will just go and throw it away. Honest evaluation is needed.
I hope some of these thoughts help. It is a terrible situation to be in.
Fr. Pat