K.: Your 5 your old is old enough to understand about death. This is actually a perfect opportunity to explain the life cycle to your child. Use whatever terms - except don't say "THE DOG WENT TO SLEEP" - that will scare her.
We had to put our guinea pig to sleep when my son was 6 - I told him that she was very very sick, the doctors are doing their best to make her better, but sometimes people and animals get so sick, that nothing the doctor does will help them get better. We have decided that instead of having Lola (our pig) suffer, we have decided to help her get to piggy heaven (or some other happy place). The doctor will give her medicine and it will help her pass away from us and go to heaven.
When I explained it this way, my son instantly said - "So Poppy will be able to take care of her?" Poppy is my dad who passed away about 3 years before - we talk about him all the time, and my son knows he is in heaven too. So I told him yes, that Poppy will take care of her. That made him feel better.
We cried ALOT that day, and for a few days after we put her down. I told my son that I was sad that Lola had to leave us, but she was in a better place and wasn't suffering anymore. He seemed okay with it and actually cried over it - which I was sort of pleased about - I'd rather he cry than keep those feelings bottled up inside (like some men we probably all know!)
You might be surprised how well your daughter handles it. You also don't want her to see everyone crying and not knowing why.
Good luck, and I'm so sorry that your parents have to put their beloved pet down.
C.