There are providers out there who do part time and do not charge a whole week.
I had to use 3 different home daycares when I was not a SAHM or a daycare mom and all 3 providers I used were okay with me putting my child /children in Mondays, Tuesdays, Fridays and just paying for those days. I wasn't using these providers simultaneously. I had to give up the first provider because we moved 20 minutes closer to my work, I gave up the second provider when I got preggo with Baby #2 and she said she was switching over to real estate by the time my 12 mo maternity would be over and I gave up provider #3 because my work schedule became flexible to the point where I could work from home and balance my work for evenings and weekends and not need daycare anymore.
Once I became a provider I also made sure that I would be open to parents who need part time so I could pay it forward. Ideally it is usually in the childs' best interest to spend as much time at their own home with their parents so if a parent can make a part time schedule work then I try to make it work. I do 3 kids part time, 1 after school only and 1 full time and it has worked well for me.
In retrospect now I would probably like to just do 2 full time children but I do appreciate the part time parents and I had 3 awesome providers who let me do that too so hubby and I could still have our child whenever we could and not go bankrupt