Decide whether an argument is necessary. In most cases, parents simply ignore your side of the story, but that is only for the time being. Maybe you have not made good decisions in the past so they are just trying to be overprotective. If they are not willing to listen right now, then pick the right time to argue.
Clear your point. If you decide to argue, make sure you have done your homework. Just asking them for something without understanding the financial or physical constraints of your family can make matters worse. Always remember how things work in your family. If you think talking only to your mother first will be helpful, then do so. However, it is important that you clear your stance by informing them about the necessary steps that you will take.
Choose your words carefully. Your tone must be polite and respectful. If you are talking in an uncustomary tone, then surely your demands will be neglected or ignored. Give them due respect and they are more likely to help you out.
Be patient. If your parents are delaying their response, so be it. Don’t just bring up the topic again and again in every family gathering. That will annoy them greatly. Give them time to think and try to acknowledge their decision even if it goes against you.
Listen to what they have to say. Initially, they may deny your requests, but don't act up on it. Ignoring them will only make matters worse. If the argument escalates, leave the room for the time being and end it by saying, ‘I’ll talk about it later, you seem in a bad mood’. That will leave an impression that you are still ready to argue but also want to respect their feelings. With time, they will understand your needs and give you the attention you want.
Sound mature and think of all the possible questions you my have to answer. This will force your parents to listen to you and take you seriously. Even if you are the youngest among your siblings, getting your facts straight and correct will leave a good impression.