Does Islam mention a caste system within marriage?

This question is motivated by a personal problem: I am in my late 20s wanting to get married to a good guy I know. We both are Bengali, Muslim; he has a good job, is on the deen, both educated to a high level. My parents first were okay with it as he was Bengali/Muslim, but once looked into it properly, they are refusing because he is apparently of a lower caste than me! (Something about fishmongers). I've tried to explain this is unislamic (more Hindu ways if anything), and I am trying to do the right thing and make my relationship Halal for us and in the eyes of Allah, but I am being told caste is Islamic? I am stuck and no longer know what to do. I am being told marrying into this caste is an unimaginable thing to do, almost degrading but I really don't see how. He and his family are good people so I'm confused about this caste issue. I would like to know: Does Islam mention a caste system within marriage and how best to approach this? Please don't judge about the relationship, rather assist me on this matter please.