Well, I'm not sure if this will work for you, I always had a crappy phone but I grew up in a time when cell phones were just starting to become an everyday, every person, thing. Perhaps explain the benefits to having a smart phone:
*First and foremost, you're not asking for the latest and greatest of smart phones, you're asking for the slightly older, yet still competitive model. This phone won't cost a fortune and you can still use it for several years without it feeling dated.
*Secondly, a smart phone acts as a GPS device and walking encyclopedia (thanks internet!). You said you have a job, soon you'll be driving if you aren't already (freshman at college or freshman at high school?). In any case, being able to use the internet on the fly not only to help yourself, but also other people (I can't tell you how many times I was asked directions at work. If I didn't know where a location was, I could only tell the person "sorry I can't help you", a smart phone with internet eliminates that)
*Thirdly, if you're willing to pay for it, you're obviously mature enough to own a phone of your choice. Maybe go over what they themselves could be saving by allowing you to pay for half or the monthly fee on your phone. If they get into the savings argument, I never saved a dime of the money I earned in high school - I blew it all on manga and pocky. I worked through college and supported myself more than fine.
Perhaps even just ask them why they're opposed to you having that phone. It would be easiest to address their concerns if we knew the root of them.
Also, I totally read your post as though you were getting a Galaxy S5. I admit I know little about iphones, so I'm not sure if my first point is true. Is there a newer iphone than the iphone 5s?