For Your Washer Example: Make Your Case
Based on your example (the washing machine), you're presenting one argument that's apparently close to your heart (the energy efficiency) but without explaining the reason (energy savings, or environmental impact).
You also don't state why your parents oppose. It may be that a new washing machine is too expensive in their eyes, or that they are more of the mind that some appliances are "for life", for their return-on-investment to be meaningful, and they see replacing one as a failure.
So, show your business acumen and flex that university-brain of yours:
Explain why it's good for the environment and why it matters for you, them, and their future grand-kids,
Explain why a more efficient will save money, and be specific: how many kwh will be saved by the new washer, and how much money will that save them based on the cost of electricity in your area? Then show how long it will take for the cost of the washer to be amortized.
Explain that the new washer may save them time, and clean better. It may also require less detergent than the older one.
Research the price you could get for the old washer if you re-sell it. While you may not want to put that old energy-hungry back on the market for someone else, there's also value in a circular economy. Donate it, recycle it, or sell it. If you can sell it, make sure to include that in your case.
For Your Point About Recurring Arguments
Inspect why these happen. Arguments happen because people are willing to argue. This may not be a bad thing.
I'm at university age so my parents can no longer influence my values, and I should have my own views on social issues by now.
For sure, and your parents surely respect that - even if they don't show or even realize it.
But you should also realize that it's not because you have your own values and opinions that theirs are suddenly invalid.
Some arguments are mostly fact-based, and can be taken out of context and analyzed in a vacuum. Many aren't, and the analysis is tainted by our experience, our culture, and our current setting. They may be your parents, and you share part of a culture and a bond, but they have a different experience, upbringing, and context than you do. You have different concerns in life, and different priorities.
Pick your Battles
On the other hand, we cannot avoid talking about these because they all start from casual topics (like replacing old appliances or my plan for the future).
That's how all arguments start. Make sure to detect when that happens, pause, and reflect on whether or not that battle is necessary.
There are usually 3 questions for that:
Can that battle be won?
If victory is what matters and it can't be won, then drop it. On the other hand, if just standing up for a conviction is good enough, even if you can't win, then that fight is still an option...
Should that battle be won? Should that battle even be fought?
You may win that battle, or you may not but have made strong points. But at what cost? Sometimes, winning an argument is not just a matter of us going "you see?" and the other party graciously accepting your winning argument with a nice "Oh, of course, I see how you were right". Most of the time, there are emotions involved, and possibly also pride.
As you're in university, maybe take a look at some Change Management modules if you have some. These will seem boring, theoretical, and contrite. And yet they contain very valuable lessons on handling personal and professional relationships, and the cost involved.
Right now, you are dealing with a lot of change, and that triggers a lot of emotions for both you and your parents, without even realizing it. You standing up to them is probably something that's already hard to deal with, as it's a nice that times are changing. And we're supremely resistant to change, and we'll want to do what we can to prevent it from happening, sometimes even if that change is beneficial.
The time we can spend together is limited (I'm attending university and doing internships), so endless arguments are really the last thing I want.
Then I'd suggest to move the cursor on the "should that battle even be fought" a bit farther from the "YES" end, and to be mindful of the time you spend with your family. Dive in on some topics, let go of others, and don't turn it up to 11 every chance you get. When things get heated, take a step.