IMO Emma has deluded herself about Harriet’s status. I think it makes it more comfortable in Emma’s mind for them to be friends but also justifies her efforts to match Harriet to a gentleman. Harriet goes from parlor boarder to the daughter of a gentleman of fortune between chapters 3 and 8 because Emma was obliged to fancy what she liked about Harriet’s parents as the girl has no idea herself. Emma is shaping Harriet in life as she painted her portrait—she embellishes Harriet’s past just as she added elegance to Harriet’s figure by making her taller in the portrait. Emma also thinks since she has noticed Harriet and introduced her into their society everyone must think of her friend as she does—I think it was quite a blow to Emma when Mr. Knightley did not in Chapter 8. See below:

Harriet Smith was the natural daughter of somebody. Somebody had placed her, several years back, at Mrs. Goddard's school, and somebody had lately raised her from the condition of scholar to that of parlour-boarder. This was all that was generally known of her history. She had no visible friends but what had been acquired at Highbury, and was now just returned from a long visit in the country to some young ladies who had been at school there with her. …She was a very pretty girl, and her beauty happened to be of a sort which Emma particularly admired. She was short, plump and fair, with a fine bloom, blue eyes, light hair, regular features, and a look of great sweetness; and before the end of the evening, Emma was as much pleased with her manners as her person, and quite determined to continue the acquaintance. (Chapter 3)

Her first attempts at usefulness were in an endeavour to find out who were the parents; but Harriet could not tell. She was ready to tell every thing in her power, but on this subject questions were vain. Emma was obliged to fancy what she liked -- but she could never believe that in the same situation she should not have discovered the truth. Harriet had no penetration. She had been satisfied to hear and believe just what Mrs. Goddard chose to tell her; and looked no farther. (Chapter 4)