Your girlfriend's parents may seem old-fashioned to you, and I realize that may make you dismiss what they say. But they have good reasons for their disapproval, and I hope you'll at least take time to consider them.
What they say is true: God does not approve of sexual relations outside of the vow of marriage. In fact, the Bible teaches us that "The body is not meant for sexual immorality" and commands us to "Flee from sexual immorality" (1 Corinthians 6:13,18). Sexual immorality was even more common in the 1st century than it is today -- but that didn't change God's command.
But why did God put the "fence" of marriage around His gift of sex? One reason is because He meant for it to be a sign of commitment -- the commitment of a husband and wife to each other. That sign, however, can only be fulfilled within the commitment of marriage. Your present relationship may seem happy for a time, but it will always lack one thing: commitment. Without the commitment of marriage, your relationship will always be insecure.
There is a larger issue, however, that I hope you'll face -- and that is the place God should have in your life. Right now you are living only for yourself. But God created you, and He loves you and wants you to know Him personally. This is the greatest discovery you will ever make -- and I pray you will make it by turning to Jesus Christ and committing your life to Him.