The parents are responsible for the kids, call the police on the parents for the kids trespassing. Tell the police you don't want them on your property anymore. The next time there's a similar problem, the parents can be held responsible. I bet that only happens once and you won't have the problem again!
I have a problem with kids playing/under my windows and also looking into and screaming into my windows (Living Room and Bed Room). The people in the rental refuse to do anything. What can I do so that I can have the privacy I so deserve?
Not much you can do. Some of these suggestions are funny but if you actually do them you'll be liable for injuries. You can't really call the police and press trespassing because it's not your property. Basically if the rental company won't do anything you might be stuck.
Check the rights of renters in your state. I doubt this is something that hasn't been addressed in the law already. You might just want to start looking for somewhere new to live for when you contract runs out.
First, put up a sign that says 'no trespassing, danger', then put thumb tacks on your window sill. E6000 glue will bond them for weeks. Let them get close enough and they will be sad they did. If you want to have fun and can plant a trip wire to an air horn... That will scare the little creepers down the street! You can get those at big hardware store and boat stores for not much money. Sound is a great deterrent... So are hurting fingers from thumb tacks!