Welcome to the forum. I am new here too.
Firstly, I will share with you my own experiences with Turkish girls and then come to a conclusion about your situation.
If I've had 10 Turkish girlfriends, only 2 of them informed their parent of my existence right after we were together. This is how it works here, unfortunately. Out of these 10 girls, I had two long-term relationships. The first one lasted 3,5 years. I was about 19 and she was 17. She was a granddaugher of a well-known businessman. I mean, when you googled him, you would see news about that guy. That was how upper class they were. My family was very average compared to hers, yet we were neighbors and ours parents knew each other. Like your situation, our relationship was a secret. Secret from her family of course. My family knew her and she used to visit us at our home. One day, checking her phone out, her family learned about us. This is what happened. They kept her at home more than a week like a prisoner and took her phone. They cut the communication between us and forced her to end the relationship. She used to love me so much, I knew she would do anything for me but I had so much respect for myself back then that I ended the relationship myself after these events. I mean she would still see me secretly after what happened, but I just couldn't take it. Now after 6-7 years, she still sends me messages sometimes on facebook and asks how I am doing. In this experience, it was not religious or conservative reasons which made her family react that way. Because they were not religious. It was class reasons.
Now the second long-term relationship is still ongoing and it's almost 4 years. This is so funny because this one is the direct opposite of the previous. She is from a poor family and she has 11 siblings. Yes 11. Her father is a farmer, mother is a housewife. They are so religious. Our relationship hasn't been kept secret from her sisters and some of her brothers. They all know about me and although they are not so warm about this relationship (because I am an atheist) they don't stand against it. But it is still a secret for her parents. They will not let their daughter to marry a non-muslim and I will have to act like a muslim when I visit them if I really want her. Again, since I have respect for myself, I will not be someone else and this relationship is a dead-end, we both know this but still carrying on. In this experience, it will be religious reason which will put us apart.
In your case, I don't think that 2 years are so much time to have the right to meet her parents, at least that is what she should be thinking. You should not force her about this matter. You should just let it slide. She wouldn't waste her time with someone she doesn't take serious. The point you should focus is what the reason of rejection could be when they know you. Nationality? I don't think so. Turkish people are not so racist. As long as you are a muslim, I don't think that they would care (if they are religious). Since you have an arab background, even if you are not a muslim, you can act like one. What looks more probable is the class rejection. Her father, as you just said is a classy man and he might want his son-in-law not to be less classy than his daughter is. If that is the case, I don't know what your background is but they would want to learn about your family. I know this is so stupid. People make choices like choosing a car. They only look at the brand, not the engine. If it is Mercedes, it is ok. One other probability of rejection is some certain group of Turkish people who hate arab people. If he belongs to that group, then you have no chance. Nevertheless, if two people truly love each other, nothing can stop them. This ideology should be your guard. I mean if she ever pleads her family as an excuse, then you will understand that the love she has for you is not enough.