Are there back exercises that will help Dad recover from his stroke?
I think it is extremely important that you get your dad to the intensive rehabilitation center in the city. If his doctors think he is medically stable, then it is in his best interest to get professional help with the rehabiliation as quickly as possible. In my experience, there is no substitute for working with professional therapists several hours a day.
In the meantime, he should work constantly at using the effected arm and leg as much as possible. Simple tasks, such as trying to move pencils (not sharpened) from one side of a desk to another, or moving similar objects carefully on the floor with his effected foot, would be helpful. He should try such things for several hours a day, in addition to more strength building exercises, like trying to lift his arm over his head, trying to lift his leg off the ground, etc. However, none of this is as helpful as working with a professional therapist.
Good luck!