As Rich mentioned, an elimination diet would be more helpful. Remember, SpA is an immune-mediated disease which means it is a combination of our immune system *and* an environmental trigger. The key to management is to identify the triggers, and avoid them. Your triggers are unique to you. What works for me or someone else may not work or not be as effect for you. Our work is find our individual triggers. It's an on-going job of finding and refining this list. Also, keep in mind that as your body heals and adjusts, your tolerance for these triggers can change. Some of theories behind this include healing a leaky gut and changing the microflora balance in your gut.
For me, low starch is my principle trigger. Limiting starch intake has allowed me to dramatically reduce my medications. After following this diet for years, I now find I am able to tolerate rice and potatoes in small quantities. Pizza is still a killer. If I eat a slice, I can guarantee that 6 days later I will feel like crap.
I'm going to offer something that is opposite to conventional wisdom. Take it as you will. I'm not sure that a vegetarian diet is fundamentally a "healthy" diet. Yes, it can be but our bodies have not evolved to be vegetarians. There are many good ethical reasons to be vegetarian, e.g., animal cruelty and environmental reasons, but it is not really a good strategy for optimal health. I also agree that the Western diet includes too much animal proteins and not nearly enough fruits and vegetables. You will very hard pressed to eliminate starches and maintain a healthful vegetarian diet. I tend to avoid vegetarian dishes and restaurants. Vegetable-based proteins nearly always come with starches. This is purely a scientific argument, I make no judgement or criticism of your personal integrity. But I do challenge you to consider what you are trying to accomplish through your vegetarian diet. If it is to improve your health, I recommend challenging the conventional wisdom. If it is for ethical reasons, you should be able to find a balance where you are fueling your body with the feedstock it needs while still respecting you ethical goals.
Hope you find this helpful if a bit challenging. Good luck.