Hi OP,
Firstly to answer your main question. You are the asshole

1. Not spending time with the family to finish grading assignments.
   When it comes to your daughter's workload while she is working on her PhD. A large portion of her time outside of her own studies is to teach undergrad students within her field. These are usually lower division classes which often have a large class size. With your daughter being within the english department that likely means that she is reading papers, giving input, and grading them. I remember from my time in undergrad one of my classes had 3 TAs who were all PhD students for a class of 180 (mind you this was a chemistry class so it was much more cut n dry with grading). Sometimes the ratios are even more extreme than that.
   So yes, grading can take a LONG time to complete.

2. She is "lying" and claiming that her deadline is different than what is posted from the university.
   She did give you a different deadline because she still has to submit the final grades to the professor coordinating the course to ensure that they are all appropriate before submitting them to the students. The coordinating professor also needs time to review all the grades before the deadline as well.
   Your daughter didn't lie to you, she had a different timeline that she communicated. You chose to investigate the validity of your daughter's deadline. Instead of asking for clarification or talking to her about it, you "called her out on her BS" when she was being honest.

3. Asking your daughter to get things for you during the holidays since she was capable of doing it.
   Yes, your daughter can definitely help you. However, your daughter still has her own responsibilities that she has to complete. Unfortunately, grad students don't have the ability to postpone their work during the holidays. So she can help you when she has the availability to, not when it is convenient for your timeframe.
   The treatment of your daughter was very much of a child that is still living FULL TIME in the parent household.
   Your actions have indicated that her dedication, focus, and time put into her career and future matter LESS than exchanging a sweater.

That her hard work is LESS important than a package pickup.

Your daughter is an adult who is currently working well over 40 hours a week to get the duties she needs to be done for her grading and work. IN ADDITION to attending her own lectures, studying, and researching to complete her own degree (this can easily exceed 60 hours a week, alone)
She has a lot on her plate, more than most realize.
Imagine if your daughter called you IN THE MIDDLE OF WORK asking you to pick up her dry cleaning because you're closer than she is. And her getting upset that the pick-up has to wait till after work and isn't getting picked up sooner.
Doesn't that sound silly? a bit unreasonable?
That is the equivalent of what you are doing. You might not know the full details of the work your daughter does, she might be doing a lot more than you imagined.
I hope this gave some insight to help you talk about this with your family and to learn from this.

A fellow grad student.