It's really not a matter of unprofessional or not. Most parents won't make their child get used to a new caregiver and then switch them back. If it's a couple of weeks many families will find a family member to help even if it means leaving state. If they have no help with family, they may plan their own vacations for that time if at all possible since you can't tell them the exact date or time it will start. If you take time off for the baby, it's extremely likely you will be starting the daycare over when you are done. That's just the facts. I've heard these stories for YEARS.
The best way to handle it is to hire a helper and pay them ALL of the money you are bringing in during that time. You need to find a good person and someone that won't ruin your business by doing a bad job. That person needs to be paid enough to make up for the fact that it's so temporary. Your parents would appreciate that you will be there to direct the person and even if you take naps with your baby, you are still there.
In my case, I just went to the hospital, had family watch the daycare children and then came home and did everything by myself after. I didn't take time off. It was easier than worrying about the helper trying to steal the kids. It's a cut throat business! Helpers will often offer the parents a cheaper deal. So there is risks with that too. I never had a c-section. So not taking time off wasn't a big deal for me. The kids and I watched tv more during the first week or two and we ordered pizza some. I put my feet up and cuddled with my newborn. I just did it like any mom of several children would do it.