The fact you have already spoken with your friend that lives almost in the neighbourhood means you can relax a little in the telling in the sense that the parents won't be stuck even on short notice just the same as needing emergency backup. You have arranged care for them should you suddenly have to move and they have the choice to use it or not. Make sure the parents and your friend know that you all understand it is temporary until the parents or new caregiver decide otherwise and then it is up to them to stay, find another person etc.

Also if you are staying somewhat in the area plot where your parents actually work and you might be surprised to find some of them would follow you to the new house if it was on their way anyways.

You could start to drop some subtle hints that you are considering moving and why but in a way that doesn't make it sound urgent. Then at least your parents can't complain they weren't warned.

You can't really start advertising till you have found the house because you won't know what street area to put down. But as soon as you have your house even if it is a couple months till you move in you could start advertising then. Make it clear in the ad that you have a dayhome now and are moving to a new house and maybe mention something that will make it a better daycare than now such as the new house has a bigger play room or basement has a bathroom or a larger backyard - makes it sound like you did it with the daycare in mind.