In this scenario, the moral obligation to come out centers on the financial support he receives from his parents. I would say he is not morally obligated to come out to his parents. Will he be morally obligated to come out to every person or entity from whom he receives money for the rest of his life? Is everyone who is gay obligated to inform every employer for whom they work of that fact in order to be considered moral?
Billy’s parents likely raised him with the expectation that he would go to college, gain an employable skill and then become self-supporting. He was not raised with the expectation of having to support himself at 18 nor offered a path that would give him the skills to do so. His “job” then, is to work hard at college and gain that skill. In a sense, his parents are paying him to do so.
Just like employers fire employees for being gay, so too may Billy’s parents cut off support if he comes out, however neither the employees nor Billy are at all immoral for not coming out in order to keep their means of financial support. An employee's only moral obligation is to perform the job which their employer hired them to do. Billy's only moral obligation is to work hard at college and leave with a skill that allows him to support himself.