head start overland park is certified by the national assn for the education of young people (ages 0-8) and they might have part time. you might be surprised, your income might qualify to allow you to take your child there. other options that are "sort of" near are valley view united methodist church and pretty sure they have part time preschool option, and brookridge has part time, brookridge day school, but I can't say how expensive they are. for part time.
there is a church that is off of 87th and I think Lamar on the southeast side of that intersection that seems to have a preschool. but i can't vouch for the quality etc
there is also a day care run by Calvin Richert and his wife Caroline off 83rd street just west of metcalf and 83rd street and they sometimes allow part time. they are AWESOME though our kids have never been in their day care they teach a class/method called Discipline WIthout damage including book etc and it's a wonderful wonderful teaching process for children ...we think the world of the Richerts.
you could probably google Discipline Without Damage overland park ks and get their phone number. wonderful, smart, calm, loving folks.
hope this helps