tell your dad that washing rattie bedding in the washing machine either alone or with your stuff WILL NOT KILL EITHER THE WASHING MACHINE OR YOUR/HIS CLOTHING. the whole point of detergent is to get the clothing clean, and it works equally as well on people clothing as it does on rattie bedding. after all, when he was a kid, i bet there wasnt disposable nappies/diapers/call-them-what-you-will around, so his mother, your grandmother, would have washed them in the washing machine... same deal as washing rattie bedding.
as for using towels, go to a second hand store and buy them for a couple of bucks each, maybe even less. you will still have to wash the towels as they will stink otherwise.