My father has Alzheimer's and Parkinson. Would the state pay a family member to take care of him during the hours she is gone?

It all depends on the state where you live. To learn more about what's possible contact your local Area Agency on Aging to see what they can share or if they can point you to state programs that might pay a stipend for a family in-home care giver. To learn more about medicaide see the Money and Legal section of this site AND search for more on your state's web site. Such programs are generally a branch of the state dept of health and welfare. Often medicaide (state aid) will supplement rent at assisted living; your father will have to use all of his monthly income and there are some spend down requirements. They won't force the sale of a home or other assets that are necessary to support a spouse. I've read of some people getting a small amount - $57 per day or something like that - from the Veterans Administration. The care giver must meet some criteria for training and be certified by the VA. Veterans retirement (nursing) homes are also a possibility of he served in WW2. There should be a VA counselor within your county or state if you live some place really rural.
Best of luck!