The four of you need to sit down and have a talk. Do so on neutral ground, e. g. not your parents' home or your girlfriend's place. What is everyone's expectations for behavior? Is someone offended? Can they come to an understanding?
I am assuming this is your first serious non-Hispanic girlfriend so you didn't have the luxury of having been through this already. Your position with your parents is that you are learning as you go and you are making mistakes but you and your girlfriend are learning from them and it is never out of a lack of respect.
Also, as an aside, this may be them pushing to see if your relationship will lead to marriage. Will it? Because a ring will probably force the issue of everyone getting along.
Another aside, the threat is for them to kick you out of the house. I am assuming you are over 18. So, honestly, how bad would it be if you stopped living with them? Not necessarily living with your girlfriend, but getting your own place? Some of the dynamic here is because their threat has some teeth. If you live successfully on their own, they lose one of their biggest threats. Something to think about while budgeting.