When parents work 12 hour shifts they generally have other days off. At minimum you are paid for a full week regardless of the fact the children don't come some of the days. They are paid for a full week and you should be too. Because of their hours they are taking up a space.
Use your daily rate for the normal workday hours. Use your before school (generally 2 hours allowed) and afterschool rate (usually 2 hours allowed). Add the three together to reach a 12 hour shift rate.
Another option is to realize that a 12 hour shift is 1 1/2 times a normal 8 hour shift so charge 1 1/2 times your normally daily rate per 12 hour shift the parent works. Remember you will be serving extra meals so decide if those are included or not.
Also don't be afraid to insist child lays down upon arrival and stays on the mat quietly till you tell them time to get up (ie other kids are arriving) or make use of TV for the time between early arrival and others coming or a combination of the two. Parent should not be expecting you to entertain the child for the full 12 hour shift.