It is very common for people to get really up in arms when it comes to protecting their ability to tell themselves that they do not have a drug or alcohol problem. By you admitting you have a problem you might have triggered a feeling of guilt in them which they promptly replaced with anger and making it so you can't be around anymore.
They are far from a support group. Being offered alcohol is a temptation and obviously neither of them worry about safety when drinking. I think you will actually find a benefit in not having contact with them. As for having your mom tell you to lie to your dad, that is unhealthy behavior on her part that is then spreading to you if you engage. It is a hard position to be put in, and only you know how to truly handle that family dynamic.
Continue your recovery! That is what is most important!