The PTA often puts this together but many will only include those who have signed up to be PTA members so may not include everyone. Room parent is a good idea or just email the teacher and ask to give your email address to the other parent.

In my son's first week of K, one of the other parents gave her child a note with her phone number, the child gave it to my child and I reached out. They are now in HS and we are still close with the family.

Does your kid's class have a room parent? They typically will collect caregivers' contact info and send it out to your class. For school wide, if your school has a PTA, ask PTA if they assist with putting together a school directory.

May I ask what are the best way to get parents contact like email address through school to arrange for playdate or parties? Homeroom teacher or PTA or admin office? This is for elementary school kids.