He has had a difficult year with sepsis, a septic knee, surgery, then falling a breaking his hip which of course required another surgery. Instead of getting stronger, he seems to be getting weaker and falling for no apparent reason. He walks with a walker, and when you ask him how it happened, he doesn't know. Two days ago he fell off the edge of the bed again after reaching for a cup of coffee on the dresser. Could it be Parkinson's Disease?
How long ago was the hip surgery? It it was just a couple of months ago, your Dad could still have lingering effects of being under for the surgery. I know for myself, for every hour I am under, it takes one month for each hour before I feel normal again. And there are side effects from medicines, and falling could be one of them.
The stiff does sound like Parkinsons--FIL did that with Parkinsons, but he did have the typical tremor. I don't recall obvious tremor with my brother, but he did have the stiffness and dementia.
Worth asking about.
Also, I have had sinus issues that also had similar symptoms.