Advice for a sandwich generation regarding child discipline?

The key phrase that you wrote is "My parents live with me."
Seems to me like if it doesn't bother you under your roof, then it shouldn't bother your parents. It would be a different story if YOU lived in THEIR house.
Sorry, I'm not advocating misbehavior in a child, certainly they need to be corrected when their behavior is inappropriate. However, if this is behavior that
1) is appropriate for her age and medical condition;
2) does not put herself or anyone else in danger;
3) doesn't bother you...
then your dad needs to relax his expectations. I mean whatever happened to grandparents spoiling their grandkids behind the parents' back? The old "what happens at grandpa's stays at grandpa's" so to speak.
Not to sound harsh, but if it bothers your parents that much, they are free to seek other accommodations.
You don't have to be a sandwich. You can simply tell dad "Dad, what your complaining about isn't an issue to me, and I'm not going to turn it into one." Then change the conversation.

Just under thoughts, I would say yes, exactly to your "what tween doesn't do all these things?" You are right. And you have a special needs tween who is not living with her own parents. This makes it the more difficult to deal with. And then there's DAD!
I think what you need to do is tell Dad that you will likely disappoint him in the way you will raise your niece, but that you will do it as you see fit, to the best of your ability. I would tell him that his input further complicates a somewhat difficult situation for you.
There are going to be tough times here. There is nothing that causes angst like the house divided, and tweens and teens are masters at manipulation. A bit of chaos works to their advantage. I sure do wish you good luck.

What are the ages of your parents? Is there any physical reason they need to live with you.

I have a 30yr old Challenged nephew. I didn't raise him but he has lived up here for the last 12 yrs. My brother raised him and there were a lot of problems with him and my brother had 2 if his own who suffered because of my nephew. Their mothers attention had to go to him.

This is just my opinion, ur not going to change Dad. And your niece does not need to be exposed to his constant complaining about her. So for her sake, maybe its time for Mom and Dad to find a place of their own. This child needs to know she is loved. She needs patience. She does not need negativity.

I also have an RN in the family. If ur one, then u have a stressful job. Taking care of a special needs child takes patience and time. To me, you would have to be a superwoman to care for this child and the responsibility of 2 parents and do it without stressing out. For me, the child would be #1.

My home consists of me, my hubs, my adult son and his 10 yr old son. My 91 yr old mom lives next door but is in my house often and eats dinner with us almost every night. Even before she was old she was a critical person with little filter. When she criticizes anything regarding my son and grandson, I remind her that:
- if I want her input, I'll ask
- just because she sees something doesn't give her license to say something
- she had her chance to raise me without interference and now it's my turn

This last truth usually stops her cold.

without checking your status, how old are your parents and why are they living with you?  do they have dementia? are they disabled?  Its harder for the older generation (as we will all become that at some point) to handle change and younger children.  it just gets on their nerves.  So either keep them apart at all times or someone needs to move into an assisted living area where they have people their own age to engage with and do things.  that way there are no young children around to bug them.  its sounds harsh but you are going to wear yourself out trying to keep them apart.  good luck.

I feel for you. I feel for your child. It’s your house and your child.

Do your parents understand about your child’s special needs? Explain it to them again if more clarification is needed.

I taught my children to respect their grandparents and they did. They adored my children. They had a wonderful relationship.

Occasionally my kids did get stressed if my mom commented about them coloring outside the lines. My mom is a perfectionist!

When mom moved in I had to set boundaries regarding my children. My mom’s motto is, ‘Everything has a place and everything in its place.’

I like an orderly house as well but you’re right about kids not being orderly all of the time. I had to learn to be creative in motivating my kids to tidy up their rooms.

I learned to prioritize and pick my battles. It’s not easy raising children and caring for a parent in your home. Mom has Parkinson’s disease so nothing was ever left out that she could trip on. Their rooms were another story!

At times I would simply shut the door to my daughters rooms. I got tired of hearing mom’s comments.

Well, mom caught on. She would joke about how it was easier to close the doors to their rooms rather than look at the mess.

I told mom that I was happy about their good grades and having good friends.

They took dance lessons, gymnastics, sports, theatre, art and so on. They were busy!

They knew where everything was in the midst of the chaos. If I did try to tidy up their things and moved things around they couldn’t find anything. Hahaha

When they tidied up their rooms they were really good organizers! So I let them be.

You do not give much info in your profile as to any problems your parents may have so I am going on the assumption that the person you are caring for is your niece and not your parents.
When you were growing up how many times did you hear any version of these:
My house my rules
My way or the highway
If you don't like it you can leave
If you think you know better move out
If this is your house...Your rules.
If your parents do not like the way you are raising your niece they can move out.
If they can not tolerate the noise and a bit of disorganization they can move out.
Or if they need to live with you is there a way that you can make a portion of the house more of an In-Law suite so they have limited contact?

Now a bit of unsolicited commentary.
You say your niece has the mentality of a small child. How young? Very young children can be expected to pick up after themselves. They can be taught how to clean up the table after a meal, if not make the bed at least smooth the sheets and blankets, just to mention a few things. There is a sense of accomplishment and a bit of pride in what they have done and the praise they get. (If these are things that your niece can not do I apologize for over stepping, you know her limitations.)

My first reaction is Gramps needs to button his mouth & but out. You are doing the parenting here! And you are putting the child's needs at the heart of her care. Thank goodness for you! 💗

You touched a nerve for me!

Sometimes I just had to roll with it. Eg a friend's Mum sneaking a bit of cappuccino froth onto my baby's milk...

But sometimes I had to stand firm. My little boy needed a solid routine with regular naps & snacks to suit his body clock - otherwise mega meltdowns. In-laws thought this was indulgent & waiting for mealtimes & nighttime for sleep was better.

Endless examples of how badly he behaved (when he was both tired & hungry). I explained. My DH explained. They finally gave in a tiny bit but would provide sugary treats. He'd go manic & even bit them.

So I enrolled him in childcare instead (with regular naps & snacks) until he was older. Later when Autism Spectrum diagnosed I provided books. I still have no idea if they ever really accepted it.

My sister with a disability does things very slowly. My dad complained about that.

I know you feel. They don't understand.

The adult is dad. He doesn’t like it he can go live elsewhere.