This is a classic case of parents thinking we work for them... We don't! We work for ourselves.
If I were in your shoes I would probably let the parent know that quiet time is mandatory due to the way the daycare is set up. However, you can always tell the parent that you will insist on quiet time, but you won't nessasarily put her down for "nap" This was she has her rest and if her body needs to sleep then it will.
My kids went through a "don't want to sleep phase. It's pretty normal. Sometimes it means they aren't getting enough sleep and are so over tired their body won't settle.