Yes you can leave them in daycare and their hungry and happy meter will not drop. Now there are two options 1) is the daycare at the park. So to get there door icon, second option and second option and lastly the leave them there not text screen so you want yes (top). This daycare option is always present regardless of generation
From second generation onwards you have a second option have the parents take care of them! So the house icon first option and with the text box comes up of leave them you want the top.
So what is the difference between the two? Well at the end of the day, day care ends and either you pick them up before this happens or they have to walk home. If you leave them all day at daycare they get really dramatic and angry at you, if you do this with the parents as far as I can tell there is no drama