I beleive the reason that Edward cannot read Bellas mind is because she has a strong mind. What i mean is some people have an open mind and others have a closed mind. Bella has a mental block that no one can penetrate. Not even Aro of the Volturi. If you remember Jasper is the only one that can control Bella. But Jaspers powers work outside the mind. He can control peoples emotions which does not involve the mind. Edward can only read minds that are open and do not have a block. But later on you realize that her power is a good thing to have. In Breaking Dawn she is able to protect everyone with a shield just by her will. This is a good trait to have. I beleive that Edward and Bella were meant to meet and fall in love like it was destiny. If you think about it she is the only one that Edwards mind reading does not work on and she is also immune to Janes powers and all of the powers of the Voluturi. That tells you right there that it was destined for her to become Edwards love and to become one of the Cullen members. Bella was put on earth for Edward.
If u read Midnight sun Edward is not able to read either Charlie or Renee's mind he can get only glimpses it .So Bella is their daughter & is a combination of both. That's why Edward's not able to read Bella's mind