Q: Is there a parental app that can limit times of usage for teens?

Take the phones at night. Have the school collect the phones at the office during school hours and give them back when classes are finished.

Parents I know use an old school method called taking the phones away.    Almost all of them have reported that their kids' grades went up too.

And best of all, it's free.

I don't think an app could have all of the necessary APIs to do what you are asking. There are some parental browsers, designed to limit content, and you can go into Restrictions in settings and make a few others, like restricting access to certain apps, but generally the types of restrictions you are looking for come from the carriers, and most at a cost. I know that AT&T has a Parental Limits program for a monthly fee. I'm surprised that Sprint does not as well. I'm curious why this say it will not work with the iPhone? I would ask someone else at Sprint to see if you get a different answer about that.

You're the dad, you are paying for the phones, are you not? There shouldn't be an "battle." What you say is law. Assert it!

Sorry - and yes, I did have teenagers who are now grown up. I didn't care if they whined about it, my word was law and there were no exceptions.

3rd party apps are not allowed to control core phone functions or other apps, under the iOS.  Such a function would have to be built into the iOS by Apple.  To suggest this to Apple, post here:

Apple - iPhone - Feedback

While you're holding your breath for Apple to implement this, enforce your rules by taking the phones away permanently if they refuse or "battle" your parental authority.

Why I mention this, is the restrictions like done at AT&T are done at the back end, the Sprint network, so it shouldn't matter what kind of phone it is. I understand the phone is new to Sprint, but this shouldn't really matter. Again, I would call their customer service, and ask for a supervisor.

Also, read this about Restrictions on the phone. http://support.apple.com/kb/HT4213

EDIT:  I just spent some time on the Sprint website. That was really painful, as they do what they can to try and hide information. Even when I got close to information about their parenting controls, they then seem to send you to their forum, and wouldn't let you view answers, just questions. I did find that you seem to be able to get their Parental controls for the iPhone, so the person that you spoke with was uninformed. However, as modular747 explained, these restrictions just handle the cellular side of things. You would need to turn off your wi-fi at home to restrict that, or again, restrictions on the phone.

Whatever Sprint's parental controls consist of, they can't turn off the phone or disable WiFi connections.  They won't be able to control any non-cellular features like iMessage, FaceTime, Facebook and a multitude of other 3rd party messaging and communication services, nor using installed games and apps

There are restrictions that can be placed, but these are all full time - no time of day restrictions.  The only way to do what you stated is to take away the phones during the times you don't want them used.

This solution may have already been offered, but if the issue is preventing the ability to call/text then I can tell you what I did to solve the problem with my teens.

1. Right now, they're using iPhone 5Ss on iOS 9, but the same solution has been in place since they had iPhone 4s.
2. We're on AT&T for wireless and Verizon for Internet, though the same solution was in place when we were on Comcast Xfinity.
   1. I signed up for smart limits and set several time limits including school time restrictions and bed time restrictions.
      1. This shuts off AT&Ts data network during the restricted hours except for "allowed numbers" defined by me. I include only other family member's numbers as allowed numbers. (911 is always available)
         1. If the phone is on your wifi network, these restrictions alone will do little to solve the problem, but in combination with the next step, is very effective
   2. In my router, both on xfinity and Verizon, I have the ability via parental controls to disable the wifi on devices in my network during time periods defined by me. I have set the wifi to be disabled at the same times as the AT&T data plan.
      1. This, combined with the smart limits, effectively turns the phone into a disconnected iPod Touch during the restriction periods.
      2. Of course, if they go somewhere where there is public wifi, my home network restrictions won't apply until they get home, but home is where the problem exists so I'm ok with that.
3. As a result, I don't have to take the phones away from my kids, and I maintain control of their ability to connect. Another advantage is that when they get grounded, or any other time I choose, I can make the restrictions "all day" and they have no contact for the duration of their punishment, but still have access to their music so they can listen to depressing music and lament internally about how unfair life is.

The only remaining issue I have is that some/many gaming apps don't require a connection to function, so they could still play games while disconnected, but neither of them uses the phone for gaming very much so I can live with that.