Sometimes there may just not be enough money in a parent's budget to buy a smartphone; or they may be worried that you could run up a big bill that they'd end-up paying. But for argument's sake, let's assume that's not the case. These are the kind of things that might persuade me (it's all about preparation)...
First, get more details from your parents about their objection:
- Find out how much is "too expensive"... they probably have some concept of the cost of a smartphone, and you might be able to surprise them by showing them a product which will be much less expensive than they thought.
- Ask "Is that the only reason?"... find out if there are any other reasons for their objection, because there's no point doing lots of research only to get another objection for every objection you overcome.
Next, research thoroughly (it's more difficult for parents to object to a well-prepared case made by their beloved kids, isn't it?)
- Be clear about what spec you need (if you just want basic apps, you don't really need an expensive quad-core phone; but on the other hand, if you need a quad-core phone, you can make a good case to justify it -- eg. "I really want to learn Linux skills, and this phone will support "Ubuntu for Android... with a base station it works as a Linux PC by itself.")
- Find the best value smartphone for what you need. (Find a value smartphone which has good reviews... and if possible, select one before the current model because it will be much cheaper. For example, you might think Google Lexus 3 is a good contender. Then do your internet research. You might find a new reconditioned phone with a guarantee from a seller with a good feedback score on e-bay.
- As well as the reasons you personally want the phone, find as many advantages as possible that your parents will think are important. (For example, "It supports Google Latitude/FourSquare which lets me share my location with you... you can see where I am"; "it has tools x, y and z which will be useful for school work", "it has apps like x, y and z which will make it easier to do ...", "it will synchronise with my Google calendar, so I'll be better organised".)
- Find the best provider who gives an amazing deal. (For example, in the UK, I found that Ovivo Mobile sells a SIM card for £10, and then gives 150 minutes of calls, 250 SMS messages and 400MB data every month for free (plus a credit for the original £10). If you can show your parents that you can use the phone for free, that must be persuasive.
- If you're offering to pay bills yourself, show your parents how you propose to afford it, and how you make sure it won't cost more than expected. Present it all clearly, with all your other research, to show how serious you are.
I guess if anything's going to work, it will be this kind of plan. And if all that fails... hey... save up and buy one yourself... a couple of days of odd jobs for family and friends will probably pay for a good 2nd hand smartphone.
I hope this helps. (^_-)