Let's be real about this - families can chat happily about the same-sex relationships of friends and celebrities, but will often have a different reaction when the gay relationship is suddenly in their own family - and it is their own child who is coming out.
Your parents might initially be very confused and shocked - in the way that any of us are when something we thought was one way, turns out to be quite different. But letting those who love you come to terms with a new and significant piece of information is not the same as seeing their reaction as evidence of homophobia.
Your parents' early emotional reaction does not mean that they don't love you or that they are rejecting your choice or abandoning you. They are very likely to go through various stages of the grief processs - and you will do best if you accept their right to process a change to the script they had of your future.
Shock can be a first response, often followed by a degree of denial (where they might say that you are just 'confused', or that you 'need to talk to someone') - or they might be angry or they may start to blame themselves.
Chances are that your parents are very likely to have to work their way through a whole range of emotions. And a lot of the way they do this, and the time it will take them, will depend on their own culture, life views personality and their own marriage.
The important thing is that you try and take a step back so that you are not pulled in to a reactive position. The very last thing you want is that this turns into an argument of accusation and counter accusation about betrayal. The issue is not about betrayal, as you know, it is about your right to choose who it is that makes you happy. And similarly, your parents have the right to process the information and make sense of it in their own way and time.
So do give them time. Talk about it with them in small portions. Speak to your parents separately too if that helps.
Remind them that you love them and that you have found love with your girlfriend, and that you are happy.
We all struggle with the unexpected. Fear of the unknown is often a part of that struggle. Fortunately, the ability to adapt to change is hardwired and is part of what makes us resilient.
If, however, the situation feels very stuck and unhappy , then it would probably be helpful to seek advice from a professional who understands family dynamics and the impact of change.