Yes, there is a Kentucky state income tax - but the sales tax is only 6%. With your father being on a pension - depending on what he is drawing each month - more than likely all of that will be returned to him in refund at tax time anyway. And - depending on his age - there are many exemptions he will qualify for - as I am beginning to understand (my mil is going through this also, as she is preparing to retire).
I recommend you contact a tax specialist on KY - there is a major 'chain' that services Kentucky, too - they can tell you, based on what your father makes - what his tax will be (I assume it is the same amount always), and how he would file, etc.
Many people become concerned about the state income taxes - simply because they are not used to it. We do, however, have a much lower cost of living, lower sales tax - and certain liscences, etc. do not exist here as they do in say - Florida or Tennesee.
My father lives in Florida, originally from Kentucky, retired from Navistar in Ohio, and he has found it all equals out in the long run.
Check with a tax service - or an accountant (many around here will speak with you on the phone as a free info service) - and I believe your questions will be answered.
Also - you may check with
http://revenue.ky.gov/ - this site will give you all the tax info you need - and I believe includes a tax calculator so that you can figure it up before you make the move.
Good luck - and welcome to the state! It really is a great place to be.