My dad was diagnosed with lung cancer in May. The tumor fractured 3 vertebra which paralyzed him. Suggestions for finding him the help he needs?

If there is spinal problem there must be already some metastasis. Has the family and Dad been informed by the doctors of options for palliative or hospice care. You don't mention your Dad's age but you Mom being in a nursing home means that he is of an age. Does he wish to go through a whole lot of torturous treatment for something that may at this point be stage IV and incurable? I would explore all the options with Dad and his doctors very carefully.
We cannot know your Dad's condition, diagnosis or prognosis, general condition. I don't know what kind of pain he is dealing with. But in truth from what you say the prognosis sounds very very guarded and uncertain. It is doubtful getting chemo would allow Dad ANY reserves to participate in rehab of any kind.
I am so sorry. I think anything those of us on Forum could advise would be guessing in a complicated and dire situation we know very little about. I do not know of anyone getting chemo and entering rehab at the same time in my career as a nurse.
I would discuss with your Dad's doctors options for help for him, but I suspect he will need placement for certain.
I do think that you are going to have to utilize family leave and travel to where your Father is to at least understand fully the situation, the prognosis, and the plan of care, and how to implement same. I am so sorry. This has to be shocking and I can't imagine how helpless you must feel at present. I hope you will update us.

Where was your dad living prior to being admitted to the hospital? If the doctors are recommending chemo what options do they have or do they suggest for his care during chemo? In my experience when a patient is being released from the hospital they need to make sure the proper living and care arrangements are made before release, in this case it sounds like that would include setting up chemo and either transport for it or a cancer hospital if he qualifies where he would stay during treatment. Have you spoken with the doctors and discharge coordinator?

Your Dad is 63? He was turned down for SS disability probably because he can get SS at 62. Its only 75% but he can get it. Also, they are always turned down once. Now you need to see a SS lawyer. He does not charge for his service. Lets say it takes a year from the date Dad applied for disability to finally get it. SS will retro payment back to the date Dad applied. So he gets one lump of money. The lawyer gets 25% of this lump sum from SS. So I would try again.

You say Mom is in a NH now. Is she on Medicaid and if so, how did she get it and Dad didn't. Is 63 the correct age? Seems young for 2 people to be in NHs,

Rehabs are not skilled nursing. They primarily are physical therapy that maybe Dad is not able to do. And chemo is usually done in a hospital setting as is radiation. NHs are not equipped for this kind of care. Dad may need to be transported to an oncology unit. Seems to me, the cancer has spread to the bone. Why is chemo being suggested and how much time is it going to buy Dad?