Lots of good suggestions here. But like AFM said, it would be helpful to know your family's objections. Also, if your parents have any disposable income you will have a far easier time. Money doesn't grow on trees. Here are some thoughts with a creative spin to help get your parents over the line:
1) Its a great investment in YOU as a child. Site the skills required to maintain and refurbish a machine. You will learn how to safely diagnose and repair printed circuit boards, and how those boards work with things like switches to control things like solenoids. Site people like Jeri Elsworth (inventor), and be prepared to show them two of her many impressive youtube videos- one on her pinball machines (like LED mod on TZ), and then another on her core engineering / work stuff.
2) Its a neutral to favorable investment from a financial perspective and is LIQUID. You aren't going to lose money, and you may even make a little bit at the end. If they have any disposable income, its a place to park a little cash. You or they can sell it in a pinch to generate cash as needed.
3) Differentiates you from other children. Similar to point 1, but a key heartstring to pull and worth its own bullet. Not only will a pinball machine result in significantly less screen time (see bullet below) but it will instill a great set of skills and give you a healthy passion to pursue in your life.
4) Less screen time. If your parents are remotely concerned about your screen time, this will pull you away from screens and get you in the real world more.
5) Your home as a destination. This is HUGE for parents of teens. Parents want a place where their child's friends WANT to be. They will be able to keep better tabs on you, know you are safe, and be seen as cool parents. A pinball machine is an anchor for this outcome, and is far less expensive than a pool.
6) Find out if your mom or dad were fans of Star Trek: The Next Generation, or even Twilight Zone (or other current themes, Simpsons, Family Guy, Spider Man, etc etc). If they are, you can use this as an additional carrot to get their support. Even if you don't end up with something of the caliber of STTNG, TZ, or SM, it might get their wheels turning a bit on the possibilities.
7) Meet them in the middle. Generate some of your own cash and ask them to match you.
If you present your case along these points, and push a bit, I have no doubt you will be successful. If your parents on the other hand need to take out a loan for a machine, start saving cause you will have to go it alone.