First, stop being pushy. When I used to grocery shop with my mom, I didn't yell at her and beg her not to buy "those animals" when she went to the deli. Of course I'd rather she didn't buy those things, but if I can eat what I want, so can she. I'd be amazed if your mom ever takes you shopping again after that. If you don't want her to think you're crazy, learn when it's appropriate not to say anything.
Talk to your parents one more time - not at the table or in the store, but maybe some night before dinner. Explain why it this is important to you and that you intend to pursue it. Offer to help shop, cook and clean up - then follow through. Few busy parents will turn that down.
If they still insist on forcing you to sit at the table, sit there. Eat everything except the meat and then just sit. Ultimately, you don't have to convince them of sh*t. Just don't put meat in your mouth. In time, they'll get the message, and even if they don't agree with it, they'll have to let it go.