Is this an only child? Sounds like the typical "high expectation" parents! you may have a little of a tough time ahead of you. Apples and bananas is right! you should, when you first meet a parent let them know that the childs sched has to at least be near your sched in the daycare or things may not work out. I always ask parents if the child naps and what time. if the time is dif from our nap time I suggest to the parent that the child be slowly moved as close as possible, for hte betterment of ALL in care. you cant have one up and crying while the others are napping, not to mention you also need a little break! Sounds like they may be a little controlling, i have had dads do the very same thing to me before, doesnt feel good when you feel like you are constantly being observed and overlooked! If I where you I would let them know the child (although young yet) does need to be on same sched as the rest of the kids, just gently inform them, you will do your best in every way pos to make her days like they are at home but let them know that there are some areas you really need all the kids to be on the same page, and nap is one of them. maybe you can attempt to push the others
30 min or so toward this childs nap time and meet in the middle? Dont forget to remind the parent also: daycare can be VERY stimulating to a 12 mo old all day long as well, resulting in some changes for the childs sleep pattern at night. maybe she needs a extra little efternoon nap to help compensate the extra stimulation?