It seems a typical cultural problem here. I have come across such problems in my life when I got to know a few people from Asia who went through the same trouble as you.
On one hand, I understand your parents, because they want to make sure that your spouse is somebody with your same background, but on the other side their requests and their acts -allow me- are frankly irreasonable and questionable (especially the money offer, which I found awful).
No wonder that your gf wants more reassurances, she has probably gone through an unbelievable amount of pressure.
Apparently your family doesn't leave you with much choice. Either you follow your heart but then prepare for a long fight with your parents, or you behave as a good son and obey to them, letting them direct your life. Both choices are legitimate and acceptable, the problem lies exclusively in what is more important to you.
I think you should make your parents realize that you are not a baby anymore and that you should be left free to make your own decisions. They are free to disagree, but at least they should leave you alone without getting in your way.
About your gf, how much do you love her? Do you love her to the point to even break up with your family for her? Because this is a possible outcome, let me tell you.
You need to reflect upon all these matters before deciding what to do. In any case, best of luck and all my solidarity, dude, you are in a difficult position.