As a parent, there are three ways that a child could impress me and make me want to get them a laptop. Of course, this is a set of personal opinions and other people may look at this in different ways.
First, I would be impressed if you could lay out a good set of reasons why you actually need a laptop. My first suspicion would be that you want the laptop so that you can go in your room and play games, surf the internet, and do other nonproductive things without me knowing about it. It would help to persuade me if you could articulate reasons why having a laptop would help you to achieve goals that I approve of. This would include demonstrating to me why our current computer setup does not suffice.
Second, I would be impressed if you cared enough about getting a laptop to contribute to buying it. If you really need this laptop, are you willing to put some of your own money into buying it? Are you willing to do extra work around the house to help pay for it? It would make me much more likely to want to buy you one if you were willing to make some sacrifices to get it.
Finally, I would be more likely to buy you the laptop if you anticipated my concerns and suggested ways to alleviate them. For example, if you know that I am worried about how much gaming you do, you could offer to show me your completed homework before you take the laptop off to play. If you know that I am worried about your use of social media, you could offer to “friend” me on those social media so that I could at least have the option of monitoring your use of those media. If you know that I am concerned about your grades, offer to make your use of the laptop contingent on keeping your grades up.
All of these things would show me that you truly need a laptop and that, if I buy you a laptop, I would not be detracting from your ability to achieve the things I want for you.
If you want to persuade your parents to get you a laptop, you'll need a good reason for it.
Do you want a laptop for schoolwork or video gaming? If it's video gaming, chances are, your parents probably will not get you a laptop. If you can prove to them that in some way, this laptop that you want will help you with your school work, maybe they may want to get it for you. Perhaps this laptop can help you with your schoolwork in the way that it makes it more convenient for you to get your work done, or maybe that there is a group project in which you can take your laptop to your friend's house to work there.
Of course, wanting a laptop does not totally have to be for the purpose of using it for school. You can also just want a laptop for gaming, like I mentioned before. If so, try to emphasize what they want to here. Sure, if you do get your laptop, you will be using it for schoolwork, but you will also be able to use it for your hobbies.
Also, pricing is another thing to consider. You must see the price and performance of the laptop you want to get. It needs to be in a good price range and good performance range. If the laptop is really good but very pricey, your parents may not want to buy it for you. Likewise, if the laptop is cheap (more appealing for your parents to buy it for you), you need to make sure it is not a laptop that will break after a day of use.
With these things in mind, you can present your idea to your parents to get you a laptop. Mention what you will be using this for, emphasizing what they want to hear, and end it with the price and specifications for the performance of the computer. In my opinion, you cannot simply just say that you want a laptop in general; you need to find the reasons why and the exact laptop you want.
That being said, you cannot show them a laptop you want to get with "GAMING LAPTOP X100, etc" written all over it, and say that it will help you with your schoolwork.
Finally, come up with a proposal to your parents. For example, tell them that you will only use the laptop for gaming when you've got all your homework and studying done for the game, and that you will not use your laptop for gaming at all when there is a big test coming up. You do not need to use exactly that proposal, but you should use something similar that will make it even more appealing for them to buy it for you.
Hope I helped! Good luck getting a laptop you want!