Are your parents vegans?
You just use the same cleanliness steps you do for the family when you are making the family meal that contains fresh meat.
The night before I put out the next day's meal in a bin on the counter to defrost. Sometimes in cool water ( it defrosts more evenly) . I wash the bin I defrosted in with hot soapy water , dry it , and put it away . I wipe down the counter with bleach wipes after preparing , and anything else I might have touched ( faucets , etc).
^That is exactly what I do when I defrost and prepare meat for the family
As for the dog eating , if you are giving chicken , cut up organ meat , muscle meat ( like chop meat ) the dogs eat it pretty much the same way they do kibble. They stand there and crunch it. Mine do not drop it on the floor or make a mess or anything. Most dogs do not from what I understand. Then I wash their bowl with hot soapy water , dry , put away .
The only thing that really makes a mess on the dog's face or fur is larger bones ( recreational bones ) that they would steady with their paws and have a good long chew on. I have a baby that crawls still , so we do not do recreational raw bones for that reason..I don;t really have time to be wiping paws and cleaning the area where he was chewing. So I give him the cooked big cow shin bones ( from pet store ) or elk antlers for chewing fun.
We have 3 large dogs on raw , 3 kids 7 and under , nobody in the last 5 months has had so much as a tummy ache.
You can also point out to your parents that kibble is just as likely to have salmonella and e coli in it ..and it is more likely to get the handlers sick because they are not as concerned about dry kibble so they do not do all the washing up after feeding ( someone feeding kibble should handle it with the same care as raw) .
Any other questions , feel free to ask .everyone here is pretty helpful