I'm really sorry to hear about your dad. I can only begin to imagine how difficult that is. Do your parents live nearby?
If your management is reasonable, I would encourage you to consider telling them about your parents. You could give it to them as a heads up that you may need to take off time suddenly (which they would need to give you, under the Family Medical Leave Act).
If you don't want to take on the extra work, you could also consider telling them that you appreciate their confidence in you but you would prefer to not take on extra duties at this time as your parents are elderly and not in the best of health and so you would like to keep your job at the current responsibility level, if at all possible. This has some risk of course--- if you sense your position or the company isn't stable, they may hold it against you. However, if you are valued and they are decent people, they may look for other options to get the work done. I've seen organizations make a lot of concessions for valued individuals with tough personal situations, because they wanted to support the employee and they understood it was a difficult situation. Essentially that person's career plateaued for awhile until they were ready to pick up the reigns again. Whether you are at this type of company, only you can tell of course.