You can make a deal with someone to barter services. You can work to earn money and buy one on your own. You can contact chair manufacturers and request one to demo and review on your blog. You can go dumpster diving. You can find businesses going out of business and selling their office furniture.
I don't think you're doing yourself a favor by asking for how to be convincing, the problem you need to solve isn't being convincing, it's how to acquire the chair.
If you're still not convinced, here are some ways to get your parents: Have them sit in the chair for dinner and see if they find it as uncomfortable. Do a high quality presentation of pros/cons to you having a new chair, don't be silly make it serious. Look for a financial barter. Yard work. Chores above and beyond normal, etc.